Q: A student advisory board has 17 members. Three members are to be chosen for the positions of chair, secretary, and webmaster. What is the probability that a specific group of three members is chosen for these positions, assuming each member is equally likely to be chosen?

$$\frac{1}{17\times16\times15}$$

Question 1

A committee of 4 is to be formed from a group of 10 engineers. In how many ways can the committee be formed if the order in which the engineers are selected does not matter?

Accepted Answer

$$\frac{10!}{4!\,6!}$$

Question 2

How many different ways can the first row of a Sudoku grid be filled with the digits 1 to 9, if each digit must appear exactly once?

Accepted Answer

$$9!$$

Question 3

A password consists of 4 different digits chosen from 0 to 9. How many possible passwords are there if no digit is repeated and the order matters?

Accepted Answer

$$\frac{10!}{6!}$$

Question 4

A building contractor is planning a subdivision with 6 one-story houses, 4 two-story houses, and 2 split-level houses. In how many distinguishable ways can the 12 houses be arranged?

Accepted Answer

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

Question 5

A state’s department of transportation receives 16 bids for a project and will hire 4 companies. How many different groups of 4 companies can be selected?

Accepted Answer

$$\frac{16!}{4!\,12!}$$

Question 6

A student advisory board has 17 members. Three members are to be chosen for the positions of chair, secretary, and webmaster. What is the probability that a specific group of three members is chosen for these positions, assuming each member is equally likely to be chosen?

Accepted Answer

$$\frac{1}{17	imes16	imes15}$$

Question 7

What is the probability of being dealt 5 diamonds from a standard deck of 52 playing cards?

Accepted Answer

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

Question 8

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 contains a dangerously high level of toxin?

Accepted Answer

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

Question 9

Which of the following best describes a permutation?

Accepted Answer

An ordered arrangement of objects

Question 10

How many ways can 3 race cars finish first, second, and third in a race with 33 cars?

Accepted Answer

$$\frac{33!}{30!}$$

Question 11

Which formula would you use to determine the number of ways to arrange 8 books on a shelf if 3 are identical copies of one title?

Accepted Answer

$$\frac{8!}{3!}$$

Question 12

A lottery requires players to choose 6 numbers from 49. How many different tickets are possible?

Accepted Answer

$$\binom{49}{6}$$

Question 13

Which of the following is the correct formula for the number of combinations of n objects taken r at a time?

Accepted Answer

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

Question 14

A group of 5 friends is to be seated in a row. In how many ways can they be seated if two of them insist on sitting together?

Accepted Answer

$$4!	imes2!$$

Question 15

Which scenario requires the use of distinguishable permutations?

Accepted Answer

Arranging the letters in the word 'BALLOON'

Counting Principles in Probability: Permutations, Combinations, and Applications

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