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Counting Principles, Permutations, and Combinations

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The Fundamental Counting Principle The number of ways a series of successive things can occur is found by multiplying the number of ways in which each thing can occur.

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Example Sue goes shopping for an outfit. She buys five blouses, three skirts and two pairs of shoes. How many different outfits can she make? (5)(3)(2) = 30 Solution:

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An executive council is to be formed from a pool of 10 qualified candidates. The council will be composed of a Chairman, Vice-Chairman, Secretary and Treasurer. All 10 candidates can be appointed for any positions. (10)(9)(8)(7) = 5040 Solution: Example

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Telephone numbers in the United States begin with three-digit area codes followed by seven-digit local telephone numbers. Area codes and local telephone numbers cannot begin with 0 or 1. How many different telephone numbers are possible? Area CodeLocal Telephone Number Solution This situation involves making choices with ten groups of items. You cannot use 0 or 1 in these groups. There are only 8 choices: 2, 3, 4, 5, 6, 7, 8, or 9. You can use 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9 in these groups. There are 10 choices per group. Text Example

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Solution We use the Fundamental Counting Principle to determine the number of different telephone numbers that are possible. The total number of telephone numbers possible is 8 2 10 2 10 2 8 2 10 2 10 2 10 2 10 2 10 2 10 = 6,400,000,000. Telephone numbers in the United States begin with three-digit area codes followed by seven-digit local telephone numbers. Area codes and local telephone numbers cannot begin with 0 or 1. How many different telephone numbers are possible? There are six billion four hundred million different telephone numbers that are possible. Text Example cont.

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Permutations The number of possible permutations if r items are taken from n items is

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Example Given the letters A, B, C, D, E, F and G, how many arrangements are there of these 7 letters taken 4 at a time. Solution:

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Combinations of n Things Taken r at a Time The number of possible combinations if r items are taken from n items is

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Example In the game of bridge, each of 4 players is dealt 13 cards. How many 13-card hands can be dealt from a 52-card deck? Solution:

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Counting Principles, Permutations, and Combinations

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