The binomial coefficient makes appearances throughout mathematics. Here are a few of them.
The expression (1 + x) is called a binomial. There are many applications where a binomial is raised to a power:
The coefficients of x in the kth power of this binomial are C(n,k). Generally:
hence the name “binomial coefficient”.
Pascal’s triangle is a triangle of numbers, starting with a single number 1, and with number 1, and each number with two more numbers diagonally below it. Each internal element is the sum of the two immediately above. The kth element in the nth row is C(n,k)
The number of ways k things can be chosen from n things, where the order doesn’t matter.
The probability of choosing any particular combination of k things (again disregarding order) from a set of n things is