The factorial function is a useful function in [combinatorics](https://en.wikipedia.org/wiki/Combinatorics) for counting things such as [permutations](https://en.wikipedia.org/wiki/Permutation) as well as the definition of [Euler's number](https://en.wikipedia.org/wiki/E_(mathematical_constant)), the base of the natural logarithm, and appears in many other areas.
Notice this function satisfies the [recurrence](https://en.wikipedia.org/wiki/Recurrence_relation) n! = n × (n-1)! which is a particularly useful viewpoint to use in [many](https://en.wikipedia.org/wiki/Gamma_function) areas of mathematics allowing the factorial to be generalized to non-integer values. (For example, this recurrence can be extended with (-1/2)! = [sqrt(π)](http://www.wolframalpha.com/input/?i=(-1%2F2)!).)
As convention, the [empty product](https://en.wikipedia.org/wiki/Empty_product), that is, the product of nothing, is usually taken to be 1, so with this definition we have 0! = 1. This convention makes sense in all the uses below.
### Uses
If you have n different objects and want to know how many ways they can be arranged in a row, there are n choices for the first object, then (after picking the first object) n-1 choices for the second object, n-2 choices for the third object, etc... and so we see there are n! ways to arrange the objects.
Another common method of counting involves [combinations](https://en.wikipedia.org/wiki/Combination) which are a given by a quotient of factorials. The combinations then come up in, for example, the [binomial formula](https://en.wikipedia.org/wiki/Binomial_theorem), the coefficients in the expansion of
#### (x + y)<sup>n</sup>
for any integer n.
Factorials also appear in many useful representations of functions, including approximations of derivatives in [Taylor's formula](https://en.wikipedia.org/wiki/Taylor%27s_theorem#Taylor's_theorem_in_one_real_variable), [exponential](https://en.wikipedia.org/wiki/Power_series#Examples) and [trigonometric](https://en.wikipedia.org/wiki/Trigonometric_functions#Series_definitions) functions.
### Computation
Computing the factorial of a positive integer is incredibly straightforward, it is simply a product of all positive integers less than or equal to itself. However, this is not an efficient approach for very large numbers, and such a product will be incredibly large as well, so it is usually better to use an approximation when looking to compute very large factorials. One simple approximation is [Stirling's approximation](https://en.wikipedia.org/wiki/Stirling%27s_approximation), namely,
