# Discrete Functions

### Permutations and Combinations

#### The Permutation Function

The permutation function is defined as:

P(n, k) =

Examples:

 P(6, 3) = = = = 6(5)(4) = 120. P(9, 2) = = = = 9(8) = 72. P(7, 1) = = = 7 P(10, 10) = = = = 10! = 3628800.

The permutation function yields the number of ways that n distinct items can be arranged in k spots. For example, P(7, 3) = = 210 . We can see that this yields the number of ways 7 items can be arranged in 3 spots -- there are 7 possibilities for the first spot, 6 for the second, and 5 for the third, for a total of 7(6)(5):

P(7, 3) = = 7(6)(5) .

Example: The coach of a basketball team is picking among 11 players for the 5 different positions in his starting lineup. How many different lineups can he pick?

P(11, 5) = = = 55440 different lineups.

#### The Combination Function

The combination function is defined as:

C(n, k) =

Examples:

 C(6, 3) = = = 20. C(9, 2) = = = 36. C(7, 1) = = = 7. C(10, 10) = = = 1.

The combination function yields the number of ways n distinct items can be chosen for k spots, when the order in which they are chosen does not matter--that is, choosing ABCDE is equivalent to choosing BAEDC. In other words, we use the combination function when all spots are equivalent.

Example: If Jim has 12 shirts, and needs to pack 7 for vacation, how many different combinations of shirts can he pack?

C(12, 7) = = 792 different combinations of shirts.

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