8.6 Counting Principles. Listing Possibilities: Ex 1 Eight pieces of paper are numbered from 1 to 8 and placed in a box. One piece of paper is drawn from.

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Presentation transcript:

8.6 Counting Principles

Listing Possibilities: Ex 1 Eight pieces of paper are numbered from 1 to 8 and placed in a box. One piece of paper is drawn from the box, its number is written down, and the piece of paper is replaced in the box. Then, a piece of paper is again drawn from the box, and its number written down. Finally, the two numbers are added together. How many different ways can a total of 12 be obtained?

Listing Possibilities: Ex 2 Eight pieces of paper are numbered from 1 to 8 and placed in a box. Two pieces of paper are drawn from the box, and the numbers on the paper are written down and totaled. How many different ways can a total of 12 be obtained?

The Fundamental Counting Principle Let E 1 and E 2 be two events. The first event E 1 can occur in m 1 different ways. After E 1 has occurred, E 2 can occur in m 2 different ways. The number of ways that the two events can occur is ways.

The Fundamental Counting Principle Ex 3 How many different pairs of letters from the English alphabet are possible?

The Fundamental Counting Principle Ex 4 How many different telephone numbers are possible within each area code? (Note: local telephone numbers cannot begin with a 0 or 1)

Permutations (When Order Matters) A permutation of n different elements such that one element is first, one is second, one is third, and so on. Ex 5: How many permutations are possible for the letters A, B, C, D, E, and F?

Permutations The number of permutations of n elements is given by Ex 6: Eight horses are running a race. In how many ways can these horses come in first, second, and third? (Assume there are no ties)

Permutations The number of permutations of n elements taken r at a time is Distinguishable Permutations: Suppose a set of n objects has n 1 of one kind of object, n 2 of a second kind, n 3 of a third kind, and so on, with n=n 1 +n 2 +…+n k. Then the number of distinguishable permutations is

Permutations: Ex 7 In how many distinguishable ways can the letters in BANANA be written?

Combinations When Order Is Not Important Ex 8: In how many different ways can three letters be chosen from the letters A, B, C, D, and E? (The order of the three letters is not important.)

Combinations The number of combinations of n elements taken r at a time is Ex 9: A standard poker hand consists of five cards dealt from a deck of 52. How many different poker hands are possible?