In 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;e17;none;1;e17;;;block;;;;no;1;139392n;36818n;;;;;eq17;comptd;;center;stack;;;;;

In general the number of permutations of n things taken r at a time is given by;e18;none;1;e18;;;block;;;;no;1;139392n;56524n;;;;;eq18;comptd;;center;stack;;;;;

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:;e19;none;1;e19;;;block;;;;no;1;139392n;81572n;;;;;eq19;comptd;;center;stack;;;;;

In general, the number of combinations of n things taken r at a time is;e20;none;1;e20;;;block;;;;no;1;139392n;100518n;;;;;eq20;comptd;;center;stack;;;;;

*The Columbia Electronic Encyclopedia,* 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.

**See more Encyclopedia articles on: **Mathematics