Q: In a horse race, a trifecta bet is won by correctly selecting the horses that finish first, second, and third, in the correct order. If there are 19 horses in the race, how many different trifecta bets are possible?

$$19 \times 18 \times 17 = 5814$$

Q: A club has 7 women and 5 men. A committee of 3 women and 2 men is to be chosen. How many different possibilities are there?

$$\dbinom{7}{3} \times \dbinom{5}{2} = 35 \times 10 = 350$$

Q: A committee of 5 is to be formed from 8 engineers and 6 scientists. How many ways can the committee be formed if it must contain at least 2 scientists?

$$\dbinom{6}{2} \times \dbinom{8}{3} + \dbinom{6}{3} \times \dbinom{8}{2} + \dbinom{6}{4} \times \dbinom{8}{1} + \dbinom{6}{5} \times \dbinom{8}{0} = 15 \times 56 + 20 \times 28 + 15 \times 8 + 6 \times 1 = 840 + 560 + 120 + 6 = 1,526$$

Q: A password consists of 4 letters followed by 2 digits. If letters and digits can be repeated, how many different passwords are possible?

$$26^4$$ \times $$10^2 = 456$$,976 \times 100 = 45,697,600$$

Question 1

The Google company must appoint three corporate officers: CEO, Executive Assistant, and COO. There are eight qualified candidates. In how many different ways can the officers be appointed?

Accepted Answer

$$8 	imes 7 	imes 6 = 336$$

Question 2

The Google company must also appoint a Planning Committee with three different members from the same pool of eight candidates. How many different ways can the committee be appointed?

Accepted Answer

$$\dbinom{8}{3} = 56$$

Question 3

In a horse race, a trifecta bet is won by correctly selecting the horses that finish first, second, and third, in the correct order. If there are 19 horses in the race, how many different trifecta bets are possible?

Accepted Answer

$$19 	imes 18 	imes 17 = 5814$$

Question 4

A bicycle shop owner has 12 mountain bicycles in the showroom. The owner wishes to select 5 of them to display at a bicycle show. How many different ways can a group of 5 be selected?

Accepted Answer

$$\dbinom{12}{5} = 792$$

Question 5

A club has 7 women and 5 men. A committee of 3 women and 2 men is to be chosen. How many different possibilities are there?

Accepted Answer

$$\dbinom{7}{3} 	imes \dbinom{5}{2} = 35 	imes 10 = 350$$

Question 6

How many different ways can a chairperson and an assistant chairperson be selected for a research project if there are seven scientists available?

Accepted Answer

$$7 	imes 6 = 42$$

Question 7

A television news director wishes to use three news stories on an evening show: one as the lead story, one as the second story, and one as the closing story. If the director has a total of eight stories to choose from, how many possible ways can the program be set up?

Accepted Answer

$$8 	imes 7 	imes 6 = 336$$

Question 8

A quality control inspector must select 4 items from a batch of 30 for inspection. In how many ways can the inspector choose the items if the order does not matter?

Accepted Answer

$$\dbinom{30}{4} = 27,405$$

Question 9

A committee of 5 is to be formed from 8 engineers and 6 scientists. How many ways can the committee be formed if it must contain at least 2 scientists?

Accepted Answer

$$\dbinom{6}{2} 	imes \dbinom{8}{3} + \dbinom{6}{3} 	imes \dbinom{8}{2} + \dbinom{6}{4} 	imes \dbinom{8}{1} + \dbinom{6}{5} 	imes \dbinom{8}{0} = 15 	imes 56 + 20 	imes 28 + 15 	imes 8 + 6 	imes 1 = 840 + 560 + 120 + 6 = 1,526$$

Question 10

A password consists of 4 letters followed by 2 digits. If letters and digits can be repeated, how many different passwords are possible?

Accepted Answer

$$26^4$$ 	imes $$10^2 = 456$$,976 	imes 100 = 45,697,600$$

Question 11

A medical researcher is testing a new drug and needs to randomly assign 5 patients to receive the drug and 7 patients to receive a placebo. In how many ways can the assignments be made?

Accepted Answer

$$\dbinom{12}{5} = 792$$

Question 12

A school is forming a debate team of 4 students from a class of 20. If two particular students refuse to be on the team together, how many different teams can be formed?

Accepted Answer

$$\dbinom{20}{4} - \dbinom{18}{2} = 4845 - 153 = 4692$$

Question 13

A company has 10 employees and wants to select a president, a vice president, and a secretary. No person can hold more than one office. How many ways can the officers be selected?

Accepted Answer

$$10 	imes 9 	imes 8 = 720$$

Question 14

A genetics lab has 6 different DNA samples. In how many ways can the samples be arranged in a line?

Accepted Answer

$$6! = 720$$

Question 15

A student must answer 5 out of 10 essay questions on an exam. In how many ways can the student choose the questions to answer?

Accepted Answer

$$\dbinom{10}{5} = 252$$

Question 16

A hospital administrator must assign 4 nurses to 4 different shifts. Each nurse can work only one shift. In how many ways can the assignments be made?

Accepted Answer

$$4! = 24$$

Question 17

A lottery requires you to select 6 numbers from 1 to 49. How many different possible lottery tickets are there?

Accepted Answer

$$\dbinom{49}{6} = 13,983,816$$

Question 18

A chemistry teacher wants to arrange 5 different chemicals in a row on a shelf. In how many different orders can the chemicals be arranged?

Accepted Answer

$$5! = 120$$

Question 19

A conference has 8 keynote speakers. In how many ways can a schedule be made if 3 speakers are to be chosen and assigned to the morning, afternoon, and evening sessions (one per session)?

Accepted Answer

$$8 	imes 7 	imes 6 = 336$$

Question 20

A biology class has 10 students. The teacher wants to divide them into two groups of 5. How many ways can this be done?

Accepted Answer

$$\dfrac{1}{2} \dbinom{10}{5} = 126$$

Combinatorics and Counting Principles in Statistics: Study Notes

스터디 가이드 - 연습 문제

학습 가이드 - 플래시카드