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)

!)!(

!),(

rrn

nrnC

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