# 7.4E: Exercises - Combinations

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Do the following problems using combinations.

 How many different 3-people committees can be chosen from ten people? How many different 5-player teams can be chosen from eight players? In how many ways can a person chose to vote for three out of five candidates on a ballot for a school board election? Compute the following: 9C2 6C4 8C3 7C4 How many 5-card hands can be chosen from a deck of cards? How many 13-card bridge hands can be chosen from a deck of cards? There are twelve people at a party. If they all shake hands, how many different hand-shakes are there? In how many ways can a student choose to do four questions out of five on a test? There are five teams in a league. How many games are played if every team plays each other twice? A team plays 15 games a season. In how many ways can it have 8 wins and 7 losses? In how many different ways can a 4-child family have 2 boys and 2 girls? A coin is tossed five times. In how many ways can it fall three heads and two tails? A team of 7 workers is assigned to a project. In how many ways can 3 of the 7 workers be selected to make a presentation to the management about their progress on the project? A real estate company has 12 houses listed for sale by their clients. In how many ways can 5 of the 12 houses be selected to be featured in an advertising brochures?

The following problems involve combinations from several different sets.

 How many 5-people committees consisting of three boys and two girls can be chosen from a group of four boys and four girls? A club has 4 men, 5 women, 8 boys and 10 girls as members. In how many ways can a group of 2 men, 3 women, 4 boys and 4 girls be chosen? How many 4-people committees chosen from 4 men and 6 women will have at least 3 men? A batch contains 10 transistors of which three are defective. If three are chosen, in how many ways can they be selected with two defective? In how many ways can five counters labeled A, B, C, D and E at a store be staffed by two men and three women chosen from a group of four men and six women? How many 4-letter word sequences consisting of two vowels and two consonants can be made from the letters of the word PHOENIX if no letter is repeated?

Three marbles are chosen from an urn that contains 5 red, 4 white, and 3 blue marbles. How many samples of the following type are possible?

 All three white. Two blue and one white One of each color. All three of the same color. At least two red. None red.

The following problems involve combinations from several different sets.

Five coins are chosen from a bag that contains 4 dimes, 5 nickels, and 6 pennies. How many samples of five coins of the following types are possible?

 At least four nickels. No pennies. Five of a kind. Four of a kind. Two of one kind and two of another kind. Three of one kind and two of another kind.

Find the number of different ways draw a 5-card hand from a deck to have the following combinations.

 Three face cards. A heart flush (all hearts). Two hearts and three diamonds Two cards of one suit, and three of another suit. Two kings and three queens. 2 cards of one value and 3 of another value

The party affiliation of the 100 United States Senators in the 114th Congress, January 2015, was:

44 Democrats, 54 Republicans, and 2 Independents.

 In how many ways could a 10 person committee be selected if it is to contain 4 Democrats, 5 Republicans, and 1 Independent? In how many different ways could a 10 person committee be selected with 6 or 7 Republicans and the Democrats (with no Independents)?

The 100 United States Senators in the 114th Congress, January 2015, included 80 men and 20 women. Suppose a committee of senators is working on legislation about wage discrimination by gender.

 In how many ways could a 12 person committee be selected to contain equal numbers of men and women. In how many ways could a 6 person committee be selected to contain fewer women than men?

This page titled 7.4E: Exercises - Combinations is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Rupinder Sekhon and Roberta Bloom via source content that was edited to the style and standards of the LibreTexts platform.