Q: A student advisory board consists of 17 members. Three members serve as chair, secretary, and webmaster. What is the probability that three specific members are chosen for these positions?

$$\dfrac{1}{17 \times 16 \times 15}$$

Q: A jury of 12 is to be selected from a pool of 40 men and 35 women. What is the probability that the jury will consist of exactly 7 men and 5 women?

$$\dfrac{\dbinom{40}{7} \times \dbinom{35}{5}}{\dbinom{75}{12}}$$

Q: How many four-digit codes can be formed from the digits 0-9 if no digit is repeated?

$$10 \times 9 \times 8 \times 7$$

Q: A password consists of 4 different letters. How many possible passwords can be formed from the 26 letters of the alphabet?

$$26 \times 25 \times 24 \times 23$$

Question 1

Which of the following best describes a permutation?

Accepted Answer

An arrangement of items where order matters and no item is repeated.

Question 2

How many different ways can the first row of a blank 9x9 Sudoku grid be filled using the digits 1 to 9, with each digit used exactly once?

Accepted Answer

$$9!$$

Question 3

A car race has 33 cars. In how many ways can the cars finish first, second, and third?

Accepted Answer

$$33 	imes 32 	imes 31$$

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

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

Question 5

Which formula gives the number of combinations of $$n$$ objects taken $$r at a $$time?

Accepted Answer

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

Question 6

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

Accepted Answer

$$\dfrac{16!}{12!4!}$$

Question 7

A student advisory board consists of 17 members. Three members serve as chair, secretary, and webmaster. What is the probability that three specific members are chosen for these positions?

Accepted Answer

$$\dfrac{1}{17 	imes 16 	imes 15}$$

Question 8

A class of 30 students will randomly select 3 students to represent the class at a competition. What is the probability that you and your two best friends are selected?

Accepted Answer

$$\dfrac{1}{\dfrac{30!}{27!3!}}$$

Question 9

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

Accepted Answer

$$\dfrac{\dbinom{13}{5}}{\dbinom{52}{5}}$$

Question 10

A jury of 12 is to be selected from a pool of 40 men and 35 women. What is the probability that the jury will consist of exactly 7 men and 5 women?

Accepted Answer

$$\dfrac{\dbinom{40}{7} 	imes \dbinom{35}{5}}{\dbinom{75}{12}}$$

Question 11

Which principle states that if one event can occur in $$m$$ ways and a second event can occur in $$n$$ ways, then the number of ways the two events can occur in sequence is $$m 	imes n$$?

Accepted Answer

Fundamental Counting Principle

Question 12

How many four-digit codes can be formed from the digits 0-9 if no digit is repeated?

Accepted Answer

$$10 	imes 9 	imes 8 	imes 7$$

Question 13

Which of the following is true about combinations?

Accepted Answer

Order does not matter and repetition is not allowed.

Question 14

If you have 5 identical red balls and 3 identical blue balls, how many distinguishable ways can you arrange all 8 balls in a row?

Accepted Answer

$$\dfrac{8!}{5!3!}$$

Question 15

Which of the following scenarios requires the use of permutations rather than combinations?

Accepted Answer

Arranging 5 books on a shelf.

Question 16

A password consists of 4 different letters. How many possible passwords can be formed from the 26 letters of the alphabet?

Accepted Answer

$$26 	imes 25 	imes 24 	imes 23$$

Question 17

Which formula would you use to find the number of ways to arrange 10 objects where 4 are identical of one type, 3 are identical of another type, and the rest are all different?

Accepted Answer

$$\dfrac{10!}{4!3!}$$

Question 18

A committee of 5 is to be chosen from 12 people. How many different committees can be formed?

Accepted Answer

$$\dfrac{12!}{7!5!}$$

Question 19

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

Accepted Answer

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

Counting Principles and Probability: Permutations, Combinations, and Applications

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