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Encyclopedia—probabilityPermutations and CombinationsIn many probability problems, sophisticated counting techniques must be used; usually this involves determining the number of permutations or combinations. The number of permutations of a set is the number of different ways in which the elements of the set can be arranged (or ordered). A set of 5 books in a row can be arranged in 120 ways, or 5×4×3×2×1=5!=120 (the symbol 5!, denoting the product of the integers from 1 to 5, is called factorial 5). If, from the five books, only three at a time are used, then the number of permutations is 60, or | P(5,3) = | 5! | = | 5x4x3x2x1 | = 60 | | (5 - 3)! | 2x1 |
In general the number of permutations of n things taken r at a time is given by
On the other hand, the number of combinations of 3 books that can be selected from 5 books refers simply to the number of different selections without regard to order. The number in this case is 10:
In general, the number of combinations of n things taken r at a time isSections in this article: The Columbia Electronic Encyclopedia, 6th ed. Copyright © 2007, Columbia University Press. All rights reserved.
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