Q: A university assigns unique four-digit codes to students, using the digits 0–9, with no digit repeated. How many possible codes can be created?

$$10 \times 9 \times 8 \times 7 = 5040$$

Q: A student advisory board has 17 members. Three members will be chosen to serve as chair, secretary, and webmaster. What is the probability that a randomly selected group of three members will be assigned to these positions?

$$\frac{1}{17 \times 16 \times 15} = \frac{1}{4080}$$

Q: A food manufacturer analyzes a sample of 400 corn kernels, 3 of which are toxic. Four kernels are randomly selected. What is the probability that exactly one kernel is toxic?

$$\frac{\binom{3}{1} \times \binom{397}{3}}{\binom{400}{4}}$$

Q: A company board has 12 members. Four distinct positions (president, vice president, secretary, treasurer) must be assigned. How many ways can these positions be filled?

$$12 \times 11 \times 10 \times 9$$

Q: In the Powerball lottery, a player must choose 5 numbers from 1 to 69 and 1 Powerball number from 1 to 26. What is the probability of winning the jackpot with one ticket?

$$\frac{1}{\binom{69}{5} \times 26}$$

Question 1

A university assigns unique four-digit codes to students, using the digits 0–9, with no digit repeated. How many possible codes can be created?

Accepted Answer

$$10 	imes 9 	imes 8 	imes 7 = 5040$$

Question 2

A building contractor plans to arrange 6 one-story houses, 4 two-story houses, and 2 split-level houses in a row. How many distinguishable arrangements are possible?

Accepted Answer

$$\frac{12!}{6!4!2!}$$

Question 3

A state park has five beaches labeled A, B, C, D, and E. If three beaches are selected for new restrooms, how many combinations are possible?

Accepted Answer

$$\frac{5!}{3!2!} = 10$$

Question 4

A student advisory board has 17 members. Three members will be chosen to serve as chair, secretary, and webmaster. What is the probability that a randomly selected group of three members will be assigned to these positions?

Accepted Answer

$$\frac{1}{17 	imes 16 	imes 15} = \frac{1}{4080}$$

Question 5

How many ways can the first row of a blank Sudoku grid (using digits 1–9, no repetition) be filled?

Accepted Answer

$$9! = 362,880$$

Question 6

A food manufacturer analyzes a sample of 400 corn kernels, 3 of which are toxic. Four kernels are randomly selected. What is the probability that exactly one kernel is toxic?

Accepted Answer

$$\frac{\binom{3}{1} 	imes \binom{397}{3}}{\binom{400}{4}}$$

Question 7

A deck of 52 playing cards contains 13 diamonds. What is the probability of being dealt 5 diamonds in a 5-card hand?

Accepted Answer

$$\frac{\binom{13}{5}}{\binom{52}{5}}$$

Question 8

A company board has 12 members. Four distinct positions (president, vice president, secretary, treasurer) must be assigned. How many ways can these positions be filled?

Accepted Answer

$$12 	imes 11 	imes 10 	imes 9$$

Question 9

A manager wants to form a three-person advisory committee from 20 employees. How many possible committees can be formed?

Accepted Answer

$$\binom{20}{3}$$

Question 10

In the Powerball lottery, a player must choose 5 numbers from 1 to 69 and 1 Powerball number from 1 to 26. What is the probability of winning the jackpot with one ticket?

Accepted Answer

$$\frac{1}{\binom{69}{5} 	imes 26}$$

Question 11

A psychologist shows a list of eight activities to a subject. How many ways can the subject pick a first, second, and third activity?

Accepted Answer

$$8 	imes 7 	imes 6 = 336$$

Question 12

Which formula gives the number of distinguishable permutations of the letters in the word 'BALLOON'?

Accepted Answer

$$\frac{7!}{1!2!2!}$$

Question 13

A jury consists of 5 men and 7 women. Three jury members are selected at random for an interview. What is the probability that all three are men?

Accepted Answer

$$\frac{\binom{5}{3}}{\binom{12}{3}}$$

Question 14

A state’s department of transportation receives 16 bids for a project and plans to hire four companies. How many different combinations of four companies can be selected?

Accepted Answer

$$\binom{16}{4}$$

Question 15

Which principle should you use to solve a problem where two or more separate events occur in sequence, and each event can occur in a certain number of ways?

Accepted Answer

Fundamental Counting Principle

Question 16

A race with 33 cars awards prizes for first, second, and third place. How many possible outcomes are there for the top three positions?

Accepted Answer

$$33 	imes 32 	imes 31$$

Question 17

Which formula gives the number of permutations of n distinct objects taken r at a time?

Accepted Answer

$$\frac{n!}{(n-r)!}$$

Question 18

A contractor wants to plant 6 oak trees, 9 maple trees, and 5 poplar trees along a street. In how many distinguishable ways can they be arranged?

Accepted Answer

$$\frac{20!}{6!9!5!}$$

Question 19

Which scenario requires the use of combinations rather than permutations?

Accepted Answer

Selecting a committee of 5 people from a group of 20

Question 20

If you want to find the number of ways to select r objects from n objects without regard to order, which formula should you use?

Accepted Answer

$$\binom{n}{r} = \frac{n!}{r!(n-r)!}$$

Permutations, Combinations, and Applications of Counting Principles

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