Q1. How many different ways can the first row of a blank 9 x 9 Sudoku grid be filled?

Background

Topic: Permutations

This question tests your understanding of how to count the number of ordered arrangements (permutations) of distinct objects.

Key formula:

Where:

• = number of distinct objects (here, 9 digits)

• = n factorial, the product of all positive integers up to n

Step-by-Step Guidance

1. Identify the number of distinct digits to fill the row: .

2. Since each digit must appear exactly once, calculate the number of ways to arrange 9 distinct digits.

3. Set up the factorial expression: .

4. Multiply out the factorial: .

Try solving on your own before revealing the answer!

Q2. Find the number of ways of forming four-digit codes in which no digit is repeated.

Background

Topic: Permutations of n objects taken r at a time

This question tests your ability to count the number of ordered arrangements when selecting a subset of objects without repetition.

Key formula:

Where:

• = total number of digits (here, 10: 0-9)

• = number of digits in the code (here, 4)

Step-by-Step Guidance

1. Identify and .

2. Set up the permutation formula: .

3. Calculate .

4. Write out and to prepare for division.

Try solving on your own before revealing the answer!

Q3. Forty-three race cars started the 2007 Daytona 500. How many ways can the cars finish first, second, and third?

Background

Topic: Permutations of n objects taken r at a time

This question tests your ability to count the number of ordered arrangements for selecting a subset of objects (race cars) for specific positions.

Key formula:

Where:

• = total number of cars (43)

• = number of positions (3: first, second, third)

Step-by-Step Guidance

1. Identify and .

2. Set up the permutation formula: .

3. Calculate .

4. Write out and to prepare for division.

Try solving on your own before revealing the answer!

Q4. In how many distinguishable ways can 6 one-story houses, 4 two-story houses, and 2 split-level houses be arranged?

Background

Topic: Distinguishable Permutations

This question tests your ability to count the number of distinguishable arrangements when some objects are identical.

Key formula:

Where:

• = total number of houses (12)

• = number of one-story houses (6)

• = number of two-story houses (4)

• = number of split-level houses (2)

Step-by-Step Guidance

1. Identify , , , .

2. Set up the distinguishable permutation formula: .

3. Calculate , , , and separately.

4. Prepare to divide by the product .

Try solving on your own before revealing the answer!

Q5. How many different combinations of four companies can be selected from 16 bidding companies?

Background

Topic: Combinations

This question tests your ability to count the number of ways to select a subset of objects without regard to order.

Key formula:

Where:

• = total number of companies (16)

• = number of companies to select (4)

Step-by-Step Guidance

1. Identify and .

2. Set up the combination formula: .

3. Calculate , , and separately.

4. Prepare to divide by the product .

Try solving on your own before revealing the answer!

Q6. A student advisory board consists of 17 members. Three members serve as the board’s chair, secretary, and webmaster. What is the probability of selecting at random the three members that hold each position?

Background

Topic: Probability using Counting Techniques (Permutations)

This question tests your ability to calculate probabilities by counting the number of favorable and possible outcomes using permutations.

Key formula:

Where:

• = total number of members (17)

• = number of positions (3)

Step-by-Step Guidance

1. Identify and .

2. Set up the permutation formula: .

3. Calculate the total number of ways to assign the positions.

4. Since there is only one favorable outcome (the specific three members), set up the probability formula.

Try solving on your own before revealing the answer!

Q7. You have 11 letters consisting of one M, four Is, four Ss, and two Ps. If the letters are randomly arranged in order, what is the probability that the arrangement spells the word Mississippi?

Background

Topic: Probability using Distinguishable Permutations

This question tests your ability to calculate probabilities by counting distinguishable arrangements when some objects are identical.

Key formula:

Where:

• = total number of letters (11)

• = number of M's (1)

• = number of I's (4)

• = number of S's (4)

• = number of P's (2)

Step-by-Step Guidance

1. Identify the counts for each letter: , , , , .

2. Set up the distinguishable permutation formula: .

3. Calculate the total number of distinguishable arrangements.

4. Since only one arrangement spells "Mississippi," set up the probability formula.

Try solving on your own before revealing the answer!

Q8. Jukebox: You look over the songs on a jukebox and determine that you like 15 of the 56 songs. (a) What is the probability that you like the next three songs that are played? (Assume a song cannot be repeated.) (b) What is the probability that you do not like the next three songs that are played? (Assume a song cannot be repeated.)

Background

Topic: Probability using Counting Techniques (Permutations)

This question tests your ability to calculate probabilities for sequential events without replacement, using permutations.

Key formula:

Where:

• = total number of songs (56)

• = number of songs played (3)

• For part (a): favorable outcomes = ways to play 3 liked songs in order

• For part (b): favorable outcomes = ways to play 3 not-liked songs in order

Step-by-Step Guidance

1. For part (a): Calculate the number of ways to play 3 liked songs in order: .

2. For part (b): Calculate the number of ways to play 3 not-liked songs in order: (since ).

3. Calculate the total number of ways to play any 3 songs in order: .

4. Set up the probability formulas for each part: .

Try solving on your own before revealing the answer!

Q9. Officers: The offices of president, vice president, secretary, and treasurer for an environmental club will be filled from a pool of 14 candidates. Six of the candidates are members of the debate team. (a) What is the probability that all of the offices are filled by members of the debate team? (b) What is the probability that none of the offices are filled by members of the debate team?

Background

Topic: Probability using Counting Techniques (Permutations)

This question tests your ability to calculate probabilities for selecting specific groups for ordered positions.

Key formula:

Where:

• = total candidates (14)

• = number of offices (4)

• For part (a): favorable outcomes =

• For part (b): favorable outcomes = (since )

Step-by-Step Guidance

1. For part (a): Calculate the number of ways to fill offices with debate team members: .

2. For part (b): Calculate the number of ways to fill offices with non-debate team members: .

3. Calculate the total number of ways to fill offices from all candidates: .

4. Set up the probability formulas for each part: .

Try solving on your own before revealing the answer!

Q10. You choose 4 people at random from a group of 1200. What is the probability that all four would rate their financial shape as excellent? (Use the pie chart for reference.)

Background

Topic: Probability using Counting Techniques (Combinations)

This question tests your ability to calculate probabilities based on proportions from a population, using combinations.

Key formula:

Where:

• = total people (1200)

• = number chosen (4)

• "Excellent" proportion = 7% (from pie chart)

• Number of "Excellent" people =

Step-by-Step Guidance

1. Calculate the number of "Excellent" people: $84$.

2. Calculate the number of ways to choose 4 "Excellent" people: .

3. Calculate the total number of ways to choose any 4 people: .

4. Set up the probability formula: .

Try solving on your own before revealing the answer!

Q11. Soil Contamination: An environmental agency is analyzing soil samples from 50 farms for lead contamination. Eight of the farms have dangerously high levels of lead. Ten farms are randomly selected from the sample. How many ways could two contaminated farms and eight noncontaminated farms be chosen?

Background

Topic: Counting Techniques (Combinations)

This question tests your ability to count the number of ways to select groups from two categories using combinations.

Key formula:

Where:

• = contaminated farms (8)

• = noncontaminated farms (42)

• Choose 2 contaminated:

• Choose 8 noncontaminated:

Step-by-Step Guidance

1. Calculate the number of ways to choose 2 contaminated farms: .

2. Calculate the number of ways to choose 8 noncontaminated farms: .

3. Multiply the two results to get the total number of ways.

Try solving on your own before revealing the answer!

Q12. Find the probability of choosing six first-shift workers from a warehouse with 24 first-shift and 17 second-shift workers, when eight workers are chosen at random.

Background

Topic: Probability using Counting Techniques (Combinations)

This question tests your ability to calculate probabilities for selecting a specific number from two categories using combinations.

Key formula:

Where:

• = first-shift workers (24)

• = second-shift workers (17)

• Choose 6 first-shift:

• Choose 2 second-shift:

• Total ways to choose 8 workers:

Step-by-Step Guidance

1. Calculate the number of ways to choose 6 first-shift workers: .

2. Calculate the number of ways to choose 2 second-shift workers: .

3. Multiply the two results to get the number of favorable outcomes.

4. Calculate the total number of ways to choose any 8 workers: .

5. Set up the probability formula: .

Try solving on your own before revealing the answer!