# Computing binomial coefficients and factorials modulo a composite number

Does anyone know of a fast algorithm to compute factorials and/or binomial coefficients in general or modulo a composite number in particular (for composite moduli I am interested in the case where the factorization is not necessarily known) I know that for primes and factorials we can apply Wilson's Theorem but I was wondering if there is also a fast method for composite numbers. For binomial coefficients I was thinking of using recursion and the addition rule: $$\binom{n}{x} + \binom{n}{x+1} = \binom{n+1}{x+1}$$ somehow but I'm not sure really how to go about it. Any suggessions? Thanks.

• Do you know the factorization of the modulus? Please edit the question to specify, as that makes a tremendous difference to the best answer.
– D.W.
Jul 21, 2014 at 22:20

If $p$ is prime and $n = n_\ell \ldots n_0$, $x = x_\ell \ldots x_0$ in base $p$, then (Lucas' theorem) $$\binom{n}{x} \equiv \binom{n_\ell}{x_\ell} \cdots \binom{n_0}{x_0} \pmod{p}.$$