binomial coefficient

Binomial coefficients are a component of the binomial theorem, which states for any elements $(x, y)$ of a commutative ring,

(x + y)^{n} = \sum_{k = 0}^{n} (\binom{n}{k}) x^{k} y^{n - k}

In the polynomial expansion of the binomial $(1 + x)^{n}$ , the coefficient of the $x^{k}$ term is called its binary coefficient, written $(\binom{n}{k})$ , said " $n$ choose $k$ ":

(\binom{n}{k}) = \frac{n!}{k! (n - k)!}

By convention, $(\binom{n}{k}) = 0$ whenever $k < 0$ or $k > n$ .

Since $n - (n - k) = k$ and multiplication is commutative, there is a symmetry in the denominator of the factorial definition; formally,

(\binom{n}{k}) = (\binom{n}{n - k})

The binomial coefficient notation is widely used in combinatorics, because of its relationship to combination: there are $(\binom{n}{k})$ ways to "choose" an unordered subset of $k$ elements from a set of $n$ elements.

05 Math/01 Encyclopedia/Pascal's rule is a recurrence relation determining elements of Pascal's triangle:

(\binom{n}{k}) = (\binom{n - 1}{k - 1}) + (\binom{n - 1}{k})