{"type":"doc","content":[{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1.1 How many registration numbers can be produced if there are no further restrictions?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Fundamental Counting Principle (FCP)"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the multiplication principle to count the number of possible outcomes when creating registration numbers, given certain restrictions on the available letters and digits."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"FCP: Multiply the number of choices for each position."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Registration number format: 3 letters followed by 2 digits."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Restrictions: Letters A and X, and digit 8 are not used."}]}]}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Let "},{"type":"inlineMath","attrs":{"latex":"L"}},{"type":"text","text":" be the number of available letters and "},{"type":"inlineMath","attrs":{"latex":"D"}},{"type":"text","text":" the number of available digits."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Count the number of available letters (excluding A and X)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Count the number of available digits (excluding 8)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For each letter position, you can use any of the available letters (repetition allowed)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For each digit position, you can use any of the available digits (repetition allowed)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the multiplication: "},{"type":"inlineMath","attrs":{"latex":"L \\times L \\times L \\times D \\times D"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: 20,250 registration numbers"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"There are 24 letters (excluding A and X) and 9 digits (excluding 8). So, "},{"type":"inlineMath","attrs":{"latex":"24 \\times 24 \\times 24 \\times 9 \\times 9 = 20,250"}},{"type":"text","text":" possible registration numbers."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1.2 How many registration numbers can be produced if vowels and consonants must alternate?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Counting with Restrictions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to count arrangements with alternating vowels and consonants, using the multiplication principle."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Vowels: E, I, O, U (A is excluded)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Consonants: All other letters except X"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Alternating pattern: e.g., vowel-consonant-vowel or consonant-vowel-consonant"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Determine the number of available vowels and consonants."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify possible alternating patterns (starting with vowel or consonant)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For each pattern, calculate the number of ways to fill each position."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply the number of choices for each position."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Add the results for each possible pattern."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: 2,304 registration numbers"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"There are 4 vowels and 20 consonants. Two patterns: vowel-consonant-vowel and consonant-vowel-consonant. Each pattern gives "},{"type":"inlineMath","attrs":{"latex":"4 \\times 20 \\times 4"}},{"type":"text","text":" or "},{"type":"inlineMath","attrs":{"latex":"20 \\times 4 \\times 20"}},{"type":"text","text":". Multiply and add: "},{"type":"inlineMath","attrs":{"latex":"320 + 1,600 = 1,920"}},{"type":"text","text":" (for letters), then multiply by "},{"type":"inlineMath","attrs":{"latex":"9 \\times 9 = 81"}},{"type":"text","text":" (for digits): "},{"type":"inlineMath","attrs":{"latex":"1,920 \\times 81 = 155,520"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2.1 How many different arrangements are possible for three married couples seated on a bench?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Permutations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of how to count the number of ways to arrange distinct objects in a line."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Permutation formula: "},{"type":"inlineMath","attrs":{"latex":"n!"}},{"type":"text","text":" for "},{"type":"inlineMath","attrs":{"latex":"n"}},{"type":"text","text":" distinct objects"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Count the total number of people (6)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the formula for arranging "},{"type":"inlineMath","attrs":{"latex":"n"}},{"type":"text","text":" distinct objects: "},{"type":"inlineMath","attrs":{"latex":"n!"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the calculation: "},{"type":"inlineMath","attrs":{"latex":"6!"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: 720 arrangements"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"There are 6 people, so "},{"type":"inlineMath","attrs":{"latex":"6! = 720"}},{"type":"text","text":" possible arrangements."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2.2 If Mr and Mrs Green must be seated in the middle, how many different arrangements are possible for the remaining persons?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Permutations with Restrictions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to count arrangements when certain positions are fixed."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Fixed positions: Treat the fixed pair as occupying specific seats."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Permutations for remaining people: "},{"type":"inlineMath","attrs":{"latex":"(n-2)!"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Fix Mr and Mrs Green in the middle two seats."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Arrange the remaining 4 people in the other seats."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate the number of ways to arrange 4 people: "},{"type":"inlineMath","attrs":{"latex":"4!"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Consider the possible arrangements of Mr and Mrs Green (they can switch places)."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: 48 arrangements"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"There are "},{"type":"inlineMath","attrs":{"latex":"4! = 24"}},{"type":"text","text":" ways to arrange the other people, and 2 ways to arrange Mr and Mrs Green in the middle. "},{"type":"inlineMath","attrs":{"latex":"24 \\times 2 = 48"}},{"type":"text","text":" arrangements."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2.3 Determine the probability that Mr and Mrs Green will sit next to each other."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Probability with Permutations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to calculate probabilities by comparing favorable arrangements to total arrangements."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Probability formula: "},{"type":"inlineMath","attrs":{"latex":"P = \\frac{\\text{favorable outcomes}}{\\text{total outcomes}}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Favorable: Mr and Mrs Green together (treat as a block)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Total: "},{"type":"inlineMath","attrs":{"latex":"6!"}},{"type":"text","text":" arrangements"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Treat Mr and Mrs Green as a single unit (block)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Count the number of ways to arrange the block and the other 4 people."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Remember to account for the two possible orders within the block."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate the total number of arrangements ("},{"type":"inlineMath","attrs":{"latex":"6!"}},{"type":"text","text":")."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the probability formula."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"\\frac{240}{720} = \\frac{1}{3}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"There are "},{"type":"inlineMath","attrs":{"latex":"5! \\times 2 = 240"}},{"type":"text","text":" favorable arrangements and "},{"type":"inlineMath","attrs":{"latex":"6! = 720"}},{"type":"text","text":" total arrangements. Probability is "},{"type":"inlineMath","attrs":{"latex":"\\frac{240}{720} = \\frac{1}{3}"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3.1 One boy and girl are a couple and want to sit next to each other at any end of the row of friends. In how many different ways can the entire group be seated?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Permutations with Restrictions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to count arrangements when a pair must sit together at either end of a row."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Couple treated as a block"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Permutations for remaining people"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Number of possible ends: 2"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Treat the couple as a block and place them at either end (2 options)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Arrange the remaining 7 people."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Count the number of ways to arrange the block (the couple can switch places)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply the number of arrangements for each step."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: 10,080 ways"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"There are 2 ends, 2 ways to arrange the couple, and "},{"type":"inlineMath","attrs":{"latex":"7! = 5040"}},{"type":"text","text":" ways to arrange the others. "},{"type":"inlineMath","attrs":{"latex":"2 \\times 2 \\times 5040 = 10,080"}},{"type":"text","text":" ways."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3.2 If all the friends are seated randomly, calculate the probability that all the girls are seated next to each other."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Probability with Permutations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to calculate the probability of a specific arrangement (all girls together) among a group."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Probability formula: "},{"type":"inlineMath","attrs":{"latex":"P = \\frac{\\text{favorable outcomes}}{\\text{total outcomes}}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Favorable: All girls together (block)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Total: "},{"type":"inlineMath","attrs":{"latex":"9!"}},{"type":"text","text":" arrangements"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Treat all girls as a single block."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Arrange the block and the boys."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Arrange the girls within the block."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate the total number of arrangements ("},{"type":"inlineMath","attrs":{"latex":"9!"}},{"type":"text","text":")."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the probability formula."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"\\frac{5! \\times 4!}{9!} = \\frac{2,880}{362,880} = 0.00794"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"There are "},{"type":"inlineMath","attrs":{"latex":"6! \\times 4! = 720 \\times 24 = 17,280"}},{"type":"text","text":" favorable arrangements and "},{"type":"inlineMath","attrs":{"latex":"9! = 362,880"}},{"type":"text","text":" total arrangements. Probability is "},{"type":"inlineMath","attrs":{"latex":"\\frac{17,280}{362,880} = 0.0476"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4.1 Nametso must choose ONE DVD from the Drama category. What is the probability that she will choose Midnight?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Probability (Simple)"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of basic probability: the chance of selecting a specific item from a group."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Probability formula: "},{"type":"inlineMath","attrs":{"latex":"P = \\frac{1}{n}"}},{"type":"text","text":" where "},{"type":"inlineMath","attrs":{"latex":"n"}},{"type":"text","text":" is the number of options"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Count the total number of drama DVDs."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the number of favorable outcomes (choosing Midnight)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the probability formula."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"\\frac{1}{5}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"There are 5 drama DVDs, so the probability is "},{"type":"inlineMath","attrs":{"latex":"\\frac{1}{5}"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4.2 How many different selections are possible if her selection must include ONE drama, ONE romance and ONE comedy?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Fundamental Counting Principle"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to use the multiplication principle to count combinations across categories."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"FCP: Multiply the number of choices for each category"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Count the number of drama DVDs."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Count the number of romance DVDs."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Count the number of comedy DVDs."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply the number of choices for each category."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: 60 selections"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"There are 5 drama, 4 romance, and 3 comedy DVDs. "},{"type":"inlineMath","attrs":{"latex":"5 \\times 4 \\times 3 = 60"}},{"type":"text","text":" possible selections."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4.3 Calculate the probability that she will have Last Hero and Laughing Dragon as part of her selection in Q4.2"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Probability with Multiple Events"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to calculate the probability of selecting specific items from multiple categories."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Probability formula: "},{"type":"inlineMath","attrs":{"latex":"P = \\frac{\\text{favorable outcomes}}{\\text{total outcomes}}"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the number of ways to select Last Hero (Drama) and Laughing Dragon (Comedy)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Count the number of ways to select a Romance DVD."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply the number of ways for each step."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate the total number of possible selections (from Q4.2)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the probability formula."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"\\frac{4}{60} = \\frac{1}{15}"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"There is 1 way to select Last Hero, 1 way to select Laughing Dragon, and 4 ways to select a Romance DVD. "},{"type":"inlineMath","attrs":{"latex":"1 \\times 4 \\times 1 = 4"}},{"type":"text","text":" favorable outcomes. Probability is "},{"type":"inlineMath","attrs":{"latex":"\\frac{4}{60} = \\frac{1}{15}"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5.1 In how many different ways can eight learners be seated?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Permutations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of arranging distinct objects in a line."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Permutation formula: "},{"type":"inlineMath","attrs":{"latex":"n!"}},{"type":"text","text":" for "},{"type":"inlineMath","attrs":{"latex":"n"}},{"type":"text","text":" distinct objects"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Count the total number of learners (8)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the formula for arranging "},{"type":"inlineMath","attrs":{"latex":"n"}},{"type":"text","text":" distinct objects: "},{"type":"inlineMath","attrs":{"latex":"n!"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the calculation: "},{"type":"inlineMath","attrs":{"latex":"8!"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: 40,320 ways"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"There are 8 learners, so "},{"type":"inlineMath","attrs":{"latex":"8! = 40,320"}},{"type":"text","text":" possible arrangements."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5.2 In how many different ways can the 8 learners be seated if 3 of the learners must sit together?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Permutations with Blocks"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to count arrangements when a subset must sit together."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Treat the 3 learners as a block"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Arrange the block and the other learners"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Arrange the learners within the block"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Treat the 3 learners as a single block."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Arrange the block and the other 5 learners: "},{"type":"inlineMath","attrs":{"latex":"6!"}},{"type":"text","text":" ways."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Arrange the 3 learners within the block: "},{"type":"inlineMath","attrs":{"latex":"3!"}},{"type":"text","text":" ways."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply the number of arrangements for each step."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: 4,320 ways"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"6! \\times 3! = 720 \\times 6 = 4,320"}},{"type":"text","text":" ways."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5.3 In how many different ways can the 8 learners be seated if 2 particular learners refuse to sit next to each other?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Permutations with Restrictions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to count arrangements with a restriction (two people must not be together)."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Total arrangements: "},{"type":"inlineMath","attrs":{"latex":"8!"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Arrangements with the two together: treat as a block"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Subtract arrangements with the two together from total"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate the total number of arrangements ("},{"type":"inlineMath","attrs":{"latex":"8!"}},{"type":"text","text":")."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate the number of arrangements where the two are together (treat as a block)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Arrange the block and the other 6 learners: "},{"type":"inlineMath","attrs":{"latex":"7!"}},{"type":"text","text":" ways."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Arrange the two within the block: "},{"type":"inlineMath","attrs":{"latex":"2!"}},{"type":"text","text":" ways."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Subtract the arrangements with the two together from the total."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: 40,320 - 10,080 = 30,240 ways"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"There are "},{"type":"inlineMath","attrs":{"latex":"8! = 40,320"}},{"type":"text","text":" total arrangements, "},{"type":"inlineMath","attrs":{"latex":"7! \\times 2! = 10,080"}},{"type":"text","text":" arrangements with the two together. "},{"type":"inlineMath","attrs":{"latex":"40,320 - 10,080 = 30,240"}},{"type":"text","text":" ways."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q6.1 Write down the total possible number of arrangements at the starting blocks for eight swimmers."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Permutations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of arranging distinct objects in a line."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Permutation formula: "},{"type":"inlineMath","attrs":{"latex":"n!"}},{"type":"text","text":" for "},{"type":"inlineMath","attrs":{"latex":"n"}},{"type":"text","text":" distinct objects"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Count the total number of swimmers (8)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the formula for arranging "},{"type":"inlineMath","attrs":{"latex":"n"}},{"type":"text","text":" distinct objects: "},{"type":"inlineMath","attrs":{"latex":"n!"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the calculation: "},{"type":"inlineMath","attrs":{"latex":"8!"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: 40,320 arrangements"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"There are 8 swimmers, so "},{"type":"inlineMath","attrs":{"latex":"8! = 40,320"}},{"type":"text","text":" possible arrangements."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q6.2 Determine the probability that Mark will be in lane 1, John in lane 2 and Michael in lane 3."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Probability with Permutations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to calculate the probability of a specific arrangement among a group."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Probability formula: "},{"type":"inlineMath","attrs":{"latex":"P = \\frac{\\text{favorable outcomes}}{\\text{total outcomes}}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Favorable: Mark in lane 1, John in lane 2, Michael in lane 3"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Total: "},{"type":"inlineMath","attrs":{"latex":"8!"}},{"type":"text","text":" arrangements"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Fix Mark, John, and Michael in lanes 1, 2, and 3."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Arrange the remaining 5 swimmers in the other lanes."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate the number of favorable arrangements ("},{"type":"inlineMath","attrs":{"latex":"5!"}},{"type":"text","text":")."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate the total number of arrangements ("},{"type":"inlineMath","attrs":{"latex":"8!"}},{"type":"text","text":")."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the probability formula."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"\\frac{120}{40,320} = 0.00298"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"There are "},{"type":"inlineMath","attrs":{"latex":"5! = 120"}},{"type":"text","text":" favorable arrangements and "},{"type":"inlineMath","attrs":{"latex":"8! = 40,320"}},{"type":"text","text":" total arrangements. Probability is "},{"type":"inlineMath","attrs":{"latex":"\\frac{120}{40,320} = 0.00298"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q8.1 How many different clients can be identified with a coding system made up of TWO letters and TWO digits?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Fundamental Counting Principle with Restrictions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to count combinations with restrictions (no repeated digits, no leading zero)."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Letters: A, B, C, D, E (may be repeated)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Digits: 0-9 (no repetition, no leading zero)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"FCP: Multiply the number of choices for each position"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Count the number of ways to choose the first letter (5 options)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Count the number of ways to choose the second letter (5 options)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Count the number of ways to choose the first digit (cannot be zero, so 9 options)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Count the number of ways to choose the second digit (cannot repeat, so 9 options including zero)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply the number of choices for each position."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: 2,025 codes"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"5 \\times 5 \\times 9 \\times 9 = 2,025"}},{"type":"text","text":" possible codes."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q8.2 Determine the least number of digits that is required for a company to uniquely identify 680,000 clients using their coding system."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Counting Principle and Minimum Requirements"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to determine the minimum number of digits needed to reach a required number of unique codes."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Letters: 5 choices for each (may be repeated)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Digits: 0-9 (no repetition, no leading zero)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"FCP: Multiply the number of choices for each position"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the formula for the number of codes: "},{"type":"inlineMath","attrs":{"latex":"5 \\times 5 \\times"}},{"type":"text","text":" (number of digit combinations)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For "},{"type":"inlineMath","attrs":{"latex":"n"}},{"type":"text","text":" digits, calculate the number of possible digit combinations (no repetition, no leading zero)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply by the number of letter combinations."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up an inequality: total codes "},{"type":"inlineMath","attrs":{"latex":"\\geq"}},{"type":"text","text":" 680,000."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve for the minimum "},{"type":"inlineMath","attrs":{"latex":"n"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: 5 digits"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"With 5 digits, "},{"type":"inlineMath","attrs":{"latex":"5 \\times 5 \\times 9 \\times 9 \\times 8 \\times 7 \\times 6 = 680,400"}},{"type":"text","text":" codes, which is just enough to identify 680,000 clients."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q9.1 In how many different ways can the clothes (7 shirts and 4 trousers) be arranged on the rail?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Permutations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of arranging distinct objects in a line."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Permutation formula: "},{"type":"inlineMath","attrs":{"latex":"n!"}},{"type":"text","text":" for "},{"type":"inlineMath","attrs":{"latex":"n"}},{"type":"text","text":" distinct objects"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Count the total number of items (7 shirts + 4 trousers = 11)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the formula for arranging "},{"type":"inlineMath","attrs":{"latex":"n"}},{"type":"text","text":" distinct objects: "},{"type":"inlineMath","attrs":{"latex":"n!"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the calculation: "},{"type":"inlineMath","attrs":{"latex":"11!"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: 39,916,800 ways"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"11! = 39,916,800"}},{"type":"text","text":" possible arrangements."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q9.2 In how many different ways can the clothes be arranged if all the shirts are to be hung next to each other and the pairs of trousers are to be hung next to each other?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Permutations with Blocks"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to count arrangements when subsets must be kept together."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Treat shirts and trousers as blocks"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Arrange the blocks"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Arrange items within each block"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Treat all shirts as a block and all trousers as a block."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Arrange the two blocks: "},{"type":"inlineMath","attrs":{"latex":"2!"}},{"type":"text","text":" ways."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Arrange the shirts within their block: "},{"type":"inlineMath","attrs":{"latex":"7!"}},{"type":"text","text":" ways."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Arrange the trousers within their block: "},{"type":"inlineMath","attrs":{"latex":"4!"}},{"type":"text","text":" ways."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply the number of arrangements for each step."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: 120,960 ways"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"2! \\times 7! \\times 4! = 2 \\times 5,040 \\times 24 = 241,920"}},{"type":"text","text":" ways."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q9.3 What is the probability that a pair of trousers will hang at the beginning of the rail and a shirt will hang at the end of the rail?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Probability with Permutations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to calculate the probability of a specific arrangement among a group."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Probability formula: "},{"type":"inlineMath","attrs":{"latex":"P = \\frac{\\text{favorable outcomes}}{\\text{total outcomes}}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Total arrangements: "},{"type":"inlineMath","attrs":{"latex":"11!"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Favorable: trousers at start, shirt at end"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Count the number of ways to place a pair of trousers at the start (4 options)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Count the number of ways to place a shirt at the end (7 options)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Arrange the remaining 9 items in the middle ("},{"type":"inlineMath","attrs":{"latex":"9!"}},{"type":"text","text":" ways)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply the number of arrangements for each step."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the probability formula."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"\\frac{4 \\times 7 \\times 9!}{11!} = \\frac{4 \\times 7 \\times 362,880}{39,916,800} = 0.254"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"There are "},{"type":"inlineMath","attrs":{"latex":"4 \\times 7 \\times 9! = 10,161,600"}},{"type":"text","text":" favorable arrangements and "},{"type":"inlineMath","attrs":{"latex":"11! = 39,916,800"}},{"type":"text","text":" total arrangements. Probability is "},{"type":"inlineMath","attrs":{"latex":"\\frac{10,161,600}{39,916,800} = 0.254"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q10.1 How many possible 4-digit combinations are there to open the lock (even-numbered, no 0 or 1, no repeated digits)?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Counting with Restrictions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to count combinations with multiple restrictions (even number, no 0 or 1, no repetition)."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Digits allowed: 2-9 (8 digits)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Last digit must be even (2, 4, 6, 8)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"No repeated digits"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"List the possible digits for each position."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For the last digit (must be even), count the options."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For the other positions, count the options (excluding used digits)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply the number of choices for each position."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: 1,680 combinations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"There are 4 options for the last digit (even), and for each, "},{"type":"inlineMath","attrs":{"latex":"7 \\times 6 \\times 5"}},{"type":"text","text":" ways to fill the other positions. "},{"type":"inlineMath","attrs":{"latex":"4 \\times 7 \\times 6 \\times 5 = 840"}},{"type":"text","text":" combinations."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q10.2 Calculate the probability that you will open the lock at the first attempt if the code is greater than 5,000 and the third digit is 2."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Probability with Multiple Restrictions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to calculate the probability of a specific arrangement with multiple restrictions."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Probability formula: "},{"type":"inlineMath","attrs":{"latex":"P = \\frac{\\text{favorable outcomes}}{\\text{total outcomes}}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Total combinations: from Q10.1"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content"