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 password consists of 3 different letters followed by 2 different digits. If no letter or digit can be repeated, how many different passwords are possible?

$$26 \times 25 \times 24 \times 10 \times 9 = 1,404,000$$

Q: A committee of 5 is to be formed from 8 men and 7 women. How many committees can be formed if the committee must have at least 2 women?

$$\dbinom{7}{2} \times \dbinom{8}{3} + \dbinom{7}{3} \times \dbinom{8}{2} + \dbinom{7}{4} \times \dbinom{8}{1} + \dbinom{7}{5} = 735 + 980 + 490 + 21 = 2,226$$

Q: A school has 10 different books and wants to arrange 4 of them on a shelf. How many different arrangements are possible?

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

Q: A medical researcher needs to assign 3 different treatments to 6 patients, with each patient receiving only one treatment and each treatment assigned to exactly 2 patients. In how many ways can this be done?

$$\dfrac{6!}{(2!\times2!\times2!)} = 90$$

Question 1

The Google company must appoint three corporate officers: CEO, Executive Assistant, and COO, from a pool of 8 qualified candidates. 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 8 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 password consists of 3 different letters followed by 2 different digits. If no letter or digit can be repeated, how many different passwords are possible?

Accepted Answer

$$26 	imes 25 	imes 24 	imes 10 	imes 9 = 1,404,000$$

Question 10

A committee of 5 is to be formed from 8 men and 7 women. How many committees can be formed if the committee must have at least 2 women?

Accepted Answer

$$\dbinom{7}{2} 	imes \dbinom{8}{3} + \dbinom{7}{3} 	imes \dbinom{8}{2} + \dbinom{7}{4} 	imes \dbinom{8}{1} + \dbinom{7}{5} = 735 + 980 + 490 + 21 = 2,226$$

Question 11

A school has 10 different books and wants to arrange 4 of them on a shelf. How many different arrangements are possible?

Accepted Answer

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

Question 12

A medical researcher needs to assign 3 different treatments to 6 patients, with each patient receiving only one treatment and each treatment assigned to exactly 2 patients. In how many ways can this be done?

Accepted Answer

$$\dfrac{6!}{(2!	imes2!	imes2!)} = 90$$

Question 13

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

Accepted Answer

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

Question 14

A team of 4 is to be chosen from 6 engineers and 5 programmers. How many teams can be formed if at least one programmer must be included?

Accepted Answer

$$\dbinom{11}{4} - \dbinom{6}{4} = 330 - 15 = 315$$

Question 15

A restaurant offers 5 appetizers, 8 main courses, and 4 desserts. If a customer orders one of each, how many different meals can be ordered?

Accepted Answer

$$5 	imes 8 	imes 4 = 160$$

Question 16

A university must assign 4 different courses to 4 different time slots. In how many ways can this be done?

Accepted Answer

$$4! = 24$$

Question 17

A deck of 52 cards is shuffled. What is the probability that the first card is an ace?

Accepted Answer

$$\dfrac{4}{52} = \dfrac{1}{13}$$

Question 18

A committee of 3 is to be chosen from 5 people. What is the probability that a particular person is on the committee?

Accepted Answer

$$\dfrac{\dbinom{4}{2}}{\dbinom{5}{3}} = \dfrac{6}{10} = 0.6$$

Combinatorics and Counting Principles in Statistics

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