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 ( n k ) {\displaystyle {\tbinom {n}{k}}} or ⁠ C ( n , k ) {\displaystyle C(n,k)} ⁠. It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n; this coefficient can be computed by the multiplicative formula ( n k ) = n × ( n − 1 ) × ⋯ × ( n − k + 1 ) k × ( k − 1 ) × ⋯ × 1 , {\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 ( n k ) = n ! k ! ( n − k ) ! . {\displaystyle {\binom {n}{k}}={\frac {n!}{k!(n-k)!}}.} For example, the fourth power of 1 + x is ( 1 + x ) 4 = ( 4 0 ) x 0 + ( 4 1 ) x 1 + ( 4 2 ) x 2 + ( 4 3 ) x 3 + ( 4 4 ) x 4 = 1 + 4 x + 6 x 2 + 4 x 3 + x 4 , {\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 ( 4 2 ) = 4 × 3 2 × 1 = 4 ! 2 ! 2 ! = 6 {\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 ( n 0 ) , ( n 1 ) , … , ( n n ) {\displaystyle {\tbinom {n}{0}},{\tbinom {n}{1}},\ldots ,{\tbinom {n}{n}}} in successive rows for n...