Counting, Permutations, & Combinations

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Presentation transcript:

Counting, Permutations, & Combinations

A counting problem asks “how many ways” some event can occur. How many three-letter codes are there using letters A, B, C, and D if no letter can be repeated? One way to solve is to list all possibilities.

ABC CAD ABD CBA ACB CDA ADB CDB ADC CBD ACD DAB BAC DAC BAD DBA BCA DCA BDA DBC BCD DCB BDC CAB 24 Ways

An experimental psychologist uses a sequence of two food rewards in an experiment regarding animal behavior. These two rewards are of three different varieties. How many different sequences of rewards are there if each variety can be used only once in each sequence? Next slide

Another way to solve is a factor tree where the number of end branches is your answer.

Fundamental Counting Principle Suppose that a certain procedure P can be broken into n successive ordered stages, S1, S2, . . . Sn, and suppose that S1 can occur in r1 ways. S2 can occur in r2 ways. Sn can occur in rn ways. Then the number of ways P can occur is

Ex. 2: An experimental psychologist uses a sequence of two food rewards in an experiment regarding animal behavior. These two rewards are of three different varieties. How many different sequences of rewards are there if each variety can be used only once in each sequence? Using the fundamental counting principle: 3 2

Permutations An r-permutation of a set of n elements is an ordered selection of r elements from the set of n elements ! means factorial Ex. 3! = 3∙2∙1 0! = 1

How many three-letter codes are there using letters A, B, C, and D if no letter can be repeated? Note: The order does matter

Combinations The number of combinations of n elements taken r at a time is Order does NOT matter! Where n & r are nonnegative integers & r < n

How many committees of three can be selected from four people? Use A, B, C, and D to represent the people Note: Does the order matter?

How many five-card hands are possible from a standard deck of cards?

In how many ways can 9 horses place 1st, 2nd, or 3rd in a race?

In how many ways can a team of 9 players be selected if there are 3 pitchers, 2 catchers, 6 in-fielders, and 4 out-fielders?