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.

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Presentation transcript:

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.

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

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

Permutations and Combinations Combinations:(order is not important)

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)

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

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

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