Binomial coefficient

In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written ${\displaystyle {\tbinom {n}{k}}.}$ It is the coefficient of the x^k term in the polynomial expansion of the binomial power (1 + x)ⁿ; this coefficient can be computed by the multiplicative formula

${\displaystyle {\binom {n}{k}}={\frac {n\times (n-1)\times \cdots \times (n-k+1)}{k\times (k-1)\times \cdots \times 1}},}$

which using factorial notation can be compactly expressed as

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

For example, the fourth power of 1 + x is

${\displaystyle {\begin{aligned}(1+x)^{4}&={\tbinom {4}{0}}x^{0}+{\tbinom {4}{1}}x^{1}+{\tbinom {4}{2}}x^{2}+{\tbinom {4}{3}}x^{3}+{\tbinom {4}{4}}x^{4}\\&=1+4x+6x^{2}+4x^{3}+x^{4},\end{aligned}}}$

and the binomial coefficient ${\displaystyle {\tbinom {4}{2}}={\tfrac {4\times 3}{2\times 1}}={\tfrac {4!}{2!2!}}=6}$ is the coefficient of the x² term.

Arranging the numbers ${\displaystyle {\tbinom {n}{0}},{\tbinom {n}{1}},\ldots ,{\tbinom {n}{n}}}$ in successive rows for n = 0, 1, 2, ... gives a triangular array called Pascal's triangle, satisfying the recurrence relation

${\displaystyle {\binom {n}{k}}={\binom {n-1}{k-1}}+{\binom {n-1}{k}}.}$

The binomial coefficients occur in many areas of mathematics, and especially in combinatorics. In combinatorics the symbol ${\displaystyle {\tbinom {n}{k}}}$ is usually read as "n choose k" because there are ${\displaystyle {\tbinom {n}{k}}}$ ways to choose an (unordered) subset of k elements from a fixed set of n elements. For example, there are ${\displaystyle {\tbinom {4}{2}}=6}$ ways to choose 2 elements from {1, 2, 3, 4}, namely {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4} and {3, 4}.

The first form of the binomial coefficients can be generalized to ${\displaystyle {\tbinom {z}{k}}}$ for any complex number z and integer k ≥ 0, and many of their properties continue to hold in this more general form.