 The gamma function's ability to generalize factorial products immediately leads to applications in many areas of mathematics; in combinatorics, and by extension in areas such as probability theory and the calculation of power series.Many expressions involving products of successive integers can be written as some combination of factorials, the most important example perhaps being that of the binomial coefficient

The example of binomial coefficients motivates why the properties of the gamma function when extended to negative numbers are natural.A binomial coefficient gives the number of ways to choose elements from a set of elements; if , there are of course no ways.