The binomial coefficient is a part of combinatorics. The binomial coefficient represent the number of possible choices of k elements out of n labelled elements (with the order of the k elements being irrelevant): that is, the number of subsets of size k in a set of size n. The binomial coefficients are written as ; they are named for their role in the expansion of the binomial expression (x+y)n.
Formulae involving binomial coefficients
Specifying a subset of size k is equivalent to specifying its complement, a subset of size n-k and vice versa. Hence
There is just one way to choose n elements out of n ("all of them") and correspondingly just one way to choose zero element ("none of them").
The number of singletons (single-element sets) is n.
The subset of size k out of n things may be split into those which do not contain the element n, which correspond to subset of n-1 of size k, and those which do contain the element n. The latter are uniquely specified by the remaining k-1 element which are drawn from the other n-1.
There are no subsets of negative size or of size greater than n.
The binomial coefficient can be used to describe the mathematics of lottery games. For example the German Lotto has a system, where you can choose 6 numbers from the numbers 1 to 49. The binomial coefficient is 13,983,816, so the probability to choose the correct six numbers is .
Binomial coefficients and prime numbers
If p is a prime number then p divides for every . The converse is also true.