Singmaster's conjecture ---------------------------
For natural numbers n and p, with p ≤ n, the binomial coefficient ( n  p )\begin{pmatrix} \,n\\\ \,p \\\ \end{pmatrix} is the number of ways of choosing p objects from n, regardless of order.
It is equal to
( n  p )=n(n−1)…(n−p+1)p!=n!(n−p)!p!\begin{pmatrix} \,n\\\ \,p \\\ \end{pmatrix}= \dfrac{n(n-1)\ldots (n-p+1)}{p!} = \dfrac{n!}{(n-p)!p!}
where n! denotes the factorial of n, namely the product of all the integers from 1 to n.