Binomial coefficient for huge n

If we want to find the binomial coefficient for a huge n(10^9), we will not be able to calculate n! in time because O(n) $\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\frac{n!}{k!(n-k)!}=\frac{n!}{(n-k)!}\frac{1}{k!}$ and transforming each of them into O(k), we obtain

Mounting example python

def naive_comb(n, k, MOD=MOD):
    assert n >= 0
    assert k >= 0
    if n < k:
        return 0
    k = min(k, n - k)
    a = 1
    b = 1
    for i in range(k):
        a *= (n - i)
        a %= MOD
        b *= (i + 1)
        b %= MOD
    return (a * mod_inverse(b, MOD)) % MOD

• For mod_inverse, see Inverse Element in mod P.

• Combination about huge n from Combination in mod P

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