5.4E: Exercises - Combinations
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Do the following problems using combinations.
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How many different 3-people committees can be chosen from ten people?
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How many different 5-player teams can be chosen from eight players?
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In how many ways can a person chose to vote for three out of five candidates on a ballot for a school board election?
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Compute the following:
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9C2
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6C4
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8C3
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7C4
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How many 5-card hands can be chosen from a deck of cards?
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How many 13-card bridge hands can be chosen from a deck of cards?
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There are twelve people at a party. If they all shake hands, how many different hand-shakes are there?
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In how many ways can a student choose to do four questions out of five on a test?
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Five points lie on a circle. How many chords can be drawn through them?
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How many diagonals does a hexagon have?
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There are five teams in a league. How many games are played if every team plays each other twice?
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A team plays 15 games a season. In how many ways can it have 8 wins and 7 losses?
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In how many different ways can a 4-child family have 2 boys and 2 girls?
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A coin is tossed five times. In how many ways can it fall three heads and two tails?
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The shopping area of a town is a square that is six blocks by six blocks. How many different routes can a taxi driver take to go from one corner of the shopping area to the opposite cater-corner?
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If the shopping area in the previous problem has a rectangular form of 5 blocks by 3 blocks, then how many different routes can a taxi driver take to drive from one end of the shopping area to the opposite kitty corner end?
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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?
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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?
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A frozen yogurt store has 9 toppings to choose from. In how many ways can 3 of the 9 toppings be selected ?
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A kindergarten teacher has 14 books about a holiday. In how many ways can she select 4 of the books to read to her class in the week before the holiday?
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The following problems involve combinations from several different sets.
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How many 5-people committees consisting of three boys and two girls can be chosen from a group of four boys and four girls?
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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?
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How many 4-people committees chosen from 4 men and 6 women will have at least 3 men?
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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?
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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?
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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?
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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?
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All three white.
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Two blue and one white
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One of each color.
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All three of the same color.
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At least two red.
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None red.
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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?
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At least four nickels.
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No pennies.
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Five of a kind.
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Four of a kind.
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Two of one kind and two of another kind.
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Three of one kind and two of another kind.
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Find the number of different ways draw a 5-card hand from a deck to have the following combinations.
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Three face cards.
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A heart flush (all hearts).
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Two hearts and three diamonds
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Two cards of one suit, and three of another suit.
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Two kings and three queens.
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2 cards of one value and 3 of another value
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The party affiliation of the 100 United States Senators in the 114
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Congress, January 2015, was:
44 Democrats, 54 Republicans, and 2 Independents.
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In how many ways could a 10 person committee be selected if it is to contain 4 Democrats, 5 Republicans, and 1 Independent?
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In how many different ways could a 10 person committee be selected with 6 or 7 Republicans and the Democrats (with no Independents)?
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The 100 United States Senators in the 114
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Congress, January 2015, included 80 men and 20 women. Suppose a committee of senators is working on legislation about wage discrimination by gender.
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In how many ways could a 12 person committee be selected to contain equal numbers of men and women.
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In how many ways could a 6 person committee be selected to contain fewer women than men?
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