Factorial - Applications

Applications

Although the factorial function has its roots in combinatorics, formulas involving factorials occur in many areas of mathematics.

• There are n! different ways of arranging n distinct objects into a sequence, the permutations of those objects.

• Often factorials appear in the denominator of a formula to account for the fact that ordering is to be ignored. A classical example is counting k-combinations (subsets of k elements) from a set with n elements. One can obtain such a combination by choosing a k-permutation: successively selecting and removing an element of the set, k times, for a total of

possibilities. This however produces the k-combinations in a particular order that one wishes to ignore; since each k-combination is obtained in k! different ways, the correct number of k-combinations is

This number is known as the binomial coefficient, because it is also the coefficient of Xk in (1 + X)n.

• Factorials occur in algebra for various reasons, such as via the already mentioned coefficients of the binomial formula, or through averaging over permutations for symmetrization of certain operations.

• Factorials also turn up in calculus; for example they occur in the denominators of the terms of Taylor's formula, where they are used as compensation terms due to the n-th derivative of xn being equivalent to n!.

• Factorials are also used extensively in probability theory.

• Factorials can be useful to facilitate expression manipulation. For instance the number of k-permutations of n can be written as

while this is inefficient as a means to compute that number, it may serve to prove a symmetry property of binomial coefficients: