Q1.1 How many registration numbers can be produced if there are no further restrictions?

Background

Topic: Fundamental Counting Principle (FCP)

This question tests your ability to count the total number of possible outcomes when forming registration numbers, given certain restrictions on which letters and digits can be used.

Key Terms and Formulas:

• FCP: Multiply the number of choices for each position.

• Letters: Exclude A and X from the alphabet (so 24 letters remain).

• Digits: Exclude 8 (so 9 digits remain: 0-7, 9).

Step-by-Step Guidance

1. Determine how many letters are available for each of the three letter positions. Remember to exclude A and X.

2. Determine how many digits are available for each digit position. Exclude the digit 8.

3. Multiply the number of choices for each letter and digit position to find the total number of registration numbers.

Try solving on your own before revealing the answer!

Q1.2 How many registration numbers can be produced if vowels and consonants must alternate?

Background

Topic: Counting with Restrictions (FCP)

This question tests your ability to count arrangements with alternating vowels and consonants, applying the multiplication principle with restrictions.

Key Terms and Formulas:

• Vowels: E, I, O, U (A is excluded).

• Consonants: All other letters except X (since X is excluded).

• Alternating pattern: Either vowel-consonant-vowel or consonant-vowel-consonant.

Step-by-Step Guidance

1. Count the number of vowels and consonants available, given the exclusions.

2. Determine the possible alternating patterns for the three letters (start with vowel or consonant).

3. For each pattern, calculate the number of ways to fill each position with the appropriate type of letter.

4. Multiply by the number of ways to choose the digits (as in Q1.1).

Try solving on your own before revealing the answer!

Q2.1 How many different arrangements are possible for three married couples seated on a bench?

Background

Topic: Permutations

This question tests your understanding of how to count the number of ways to arrange a group of people in a line.

Key Terms and Formulas:

• Permutation: The number of ways to arrange n distinct objects is .

Step-by-Step Guidance

1. Count the total number of people to be arranged (three couples = six people).

2. Use the permutation formula to calculate the number of ways to arrange six people in a row.

Try solving on your own before revealing the answer!

Q2.2 If Mr and Mrs Green must be seated in the middle, how many different arrangements are possible for the remaining persons?

Background

Topic: Permutations with Restrictions

This question tests your ability to count arrangements when certain people must occupy specific positions.

Key Terms and Formulas:

• Permutation: for n distinct objects.

• Restriction: Mr and Mrs Green must be in the two middle seats (positions 3 and 4).

Step-by-Step Guidance

1. Identify the two middle seats and note that Mr and Mrs Green must occupy them. Consider the number of ways to arrange them in those two seats.

2. Arrange the remaining four people in the other four seats.

3. Multiply the number of ways to arrange Mr and Mrs Green by the number of ways to arrange the other four people.

Try solving on your own before revealing the answer!

Q2.3 Determine the probability that Mr and Mrs Green will sit next to each other.

Background

Topic: Probability with Arrangements

This question tests your ability to calculate the probability of a specific arrangement occurring among all possible arrangements.

Key Terms and Formulas:

• Probability:

• Arrangements with Mr and Mrs Green together: Treat them as a single unit, then arrange the unit and the other people.

Step-by-Step Guidance

1. Treat Mr and Mrs Green as a single "block" to ensure they are together. Count the number of ways to arrange this block and the other four people.

2. Within the block, Mr and Mrs Green can be arranged in 2 ways (Mr first or Mrs first).

3. Calculate the total number of arrangements with them together.

4. Recall the total number of arrangements from Q2.1.

5. Set up the probability formula using the numbers from the previous steps.

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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?

Background

Topic: Arrangements with a Block at the End

This question tests your ability to count arrangements when a specific pair must sit together at either end of a row.

Key Terms and Formulas:

• Permutation: for n distinct objects.

• Block arrangement: Treat the couple as a single unit, but only at the ends.

Step-by-Step Guidance

1. Consider the two possible ends where the couple can sit (left end or right end).

2. Within the couple, the boy and girl can switch places (2 ways).

3. Arrange the remaining 7 friends in the other seats.

4. Multiply the number of ways for each step to get the total arrangements.

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Q3.2 If all the friends are seated randomly, calculate the probability that all the girls are seated next to each other.

Background

Topic: Probability with Grouped Arrangements

This question tests your ability to calculate the probability that a subset of people (all girls) are seated together in a random arrangement.

Key Terms and Formulas:

• Probability:

• Block arrangement: Treat all girls as a single unit.

Step-by-Step Guidance

1. Treat the four girls as a single block, so you have 6 units to arrange (the block + 5 boys).

2. Arrange the 6 units in a row.

3. Arrange the 4 girls within their block.

4. Calculate the total number of arrangements for all 9 friends.

5. Set up the probability formula using the numbers from the previous steps.

Try solving on your own before revealing the answer!

Q4.1 Nametso must choose ONE DVD from the Drama category. What is the probability that she will choose Midnight?

Background

Topic: Basic Probability

This question tests your understanding of simple probability when choosing one item from a group.

Key Terms and Formulas:

• Probability:

Step-by-Step Guidance

1. Count the total number of Drama DVDs available.

2. Identify how many of these are "Midnight" (should be 1).

3. Set up the probability formula using these numbers.

Try solving on your own before revealing the answer!

Q4.2 How many different selections are possible if her selection must include ONE drama, ONE romance and ONE comedy?

Background

Topic: Fundamental Counting Principle (FCP)

This question tests your ability to count the number of ways to make selections from multiple categories.

Key Terms and Formulas:

• FCP: Multiply the number of choices for each category.

Step-by-Step Guidance

1. Count the number of Drama DVDs, Romance DVDs, and Comedy DVDs.

2. Multiply the number of choices in each category to find the total number of possible selections.

Try solving on your own before revealing the answer!

Q4.3 Calculate the probability that she will have Last Hero and Laughing Dragon as part of her selection in Q4.2.

Background

Topic: Probability with Multiple Selections

This question tests your ability to calculate the probability of selecting specific items from multiple categories.

Key Terms and Formulas:

• Probability:

Step-by-Step Guidance

1. Identify the categories for "Last Hero" and "Laughing Dragon" (Drama and Comedy, respectively).

2. For a favorable outcome, "Last Hero" and "Laughing Dragon" must be chosen, and any Romance DVD can be chosen.

3. Count the number of favorable outcomes (number of Romance DVDs).

4. Recall the total number of possible selections from Q4.2.

5. Set up the probability formula using these numbers.

Try solving on your own before revealing the answer!

Q5.1 In how many different ways can 8 learners be seated?

Background

Topic: Permutations

This question tests your understanding of how to count the number of ways to arrange a group of people in a line.

Key Terms and Formulas:

• Permutation: for n distinct objects.

Step-by-Step Guidance

1. Count the total number of learners to be arranged (8).

2. Use the permutation formula to calculate the number of ways to arrange 8 people in a row.

Try solving on your own before revealing the answer!