Finding binomial coefficient for large n and k modulo m

Question

I want to compute nCk mod m with following constraints:

n<=10^18

k<=10^5

m=10^9+7

I have read this article:

Calculating Binomia

Answer 1

We want to compute nCk (mod p). I'll handle when 0 <= k <= p-2, because Lucas's theorem handles the rest.

Wilson's theorem states that for prime p, (p-1)! = -1 (mod p), or equivalently (p-2)! = 1 (mod p) (by division).

By division: (k!)^(-1) = (p-2)!/(k!) = (p-2)(p-3)...(k+1) (mod p)

Thus, the binomial coefficient is n!/(k!(n-k)!) = n(n-1)...(n-k+1)/(k!) = n(n-1)...(n-k+1)(p-2)(p-3)...(k+1) (mod p)

Voila. You don't have to do any inverse computations or anything like that. It's also fairly easy to code. A couple optimizations to consider: (1) you can replace (p-2)(p-3)... with (-2)(-3)...; (2) nCk is symmetric in the sense that nCk = nC(n-k) so choose the half that requires you to do less computations.