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Permutations and Combinations

Permutations and Combinations. Independent Events: Events that do not affect each other Spinning a number 6 and then spinning a number 5 on the same spinner. Picking an ace out of a deck of cards and rolling a di and getting a 6

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Permutations and Combinations

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  1. Permutations and Combinations • Independent Events: • Events that do not affect each other • Spinning a number 6 and then spinning a number 5 on the same spinner. • Picking an ace out of a deck of cards and rolling a di and getting a 6 • Picking an ace of hearts out of a deck of cards, then picking a jack of clubs out of the same deck with replacement. • Dependent Event • Events that affect each other. • Picking an ace of hearts out of a deck of cards, then picking a jack of clubs out of the same deck without replacement.

  2. Permutations and Combinations • Basic Counting Principle: • When one event can be chosen p ways and another event can be chosen q ways, then the events can have p*q ways • Example: How many ways can Mr. Vogle make an ugly outfit if he has 5 pairs of ugly pants, 6 ugly shirts and 2 pairs of shoes? • 5*6*2=60

  3. Permutations and Combinations • Permutations: • P(n,n)=n! (permutation of n things taken n at a time) • P(n,r)= n!/(n-r)! • Order matters • Example: How many ways can you arrange 5 subject books? • P(5,5) • =120

  4. Permutations and Combinations • Combinations:(order is not important)

  5. Permutations and Combinations • How many ways can you order 4 different class officers from a class council of 12 people? (Does order matter) • P(12,4) • How many ways can you order a 4 person sub committee from a class council of 12 people? • C(12,4)

  6. Permutations and Combinations • At Kennedy High School there are 12 names on the ballot for sophomore class officers. Six will be selected to form a class committee. • a. How many different committees of 6 can be formed? • b. In how many ways can a committee of 6 be formed if each student has a different responsibility? • c. If there are 7 girls and 5 boys on the ballot, how many committees of 3 girls and 3 boys can be formed? • Answer to A • A. Order is not important in this situation, so the selection is a combination of 12 people chosen 6 at a time.C(12, 6)=924

  7. Permutations and Combinations • At Kennedy High School there are 12 names on the ballot for sophomore class officers. Six will be selected to form a class committee. • b. In how many ways can a committee of 6 be formed if each student has a different responsibility? • Answer • P(12, 6) =665,280

  8. Permutations and Combinations At Kennedy High School there are 12 names on the ballot for sophomore class officers. Six will be selected to form a class committee. • c. If there are 7 girls and 5 boys on the ballot, how many committees of 3 girls and 3 boys can be formed? • Answer • Since the events are independent, the answer is the product of the combinations C(5, 3) and C(7, 3). • C(5, 3) C(7, 3) =350

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