Binomial theorem
Formulas

The coefficient of x y is given by the formula (n k) = n ! k ! (n − k) !, {\displaystyle {\binom {n}{k}}={\frac {n!}{k!\;(n-k)!}},} which is defined in terms of the factorial function n !. Equivalently, this formula can be written (n k) = n (n − 1) ⋯ (n − k + 1) k (k − 1) ⋯ 1 = ∏ ℓ = 1 k n − ℓ + 1 ℓ = ∏ ℓ = 0 k − 1 n − ℓ k − ℓ {\displaystyle {\binom {n}{k}}={\frac {n(n-1)\cdots (n-k+1)}{k(k-1)\cdots 1}}=\prod _{\ell =1}^{k}{\frac {n-\ell +1}{\ell }}=\prod _{\ell =0}^{k-1}{\frac {n-\ell }{k-\ell }}} with k factors in both the numerator and denominator of the fraction. Although this formula involves a fraction, the binomial coefficient (n k) {\displaystyle {\tbinom {n}{k}}} is actually an integer.
oulipo:avoid_letter[e], startend:quotation, punctuation:no_comma