0830 commoncode

$$ \frac{(a+b+c+\cdots+z)!}{a!b!c!\cdots z!} \pmod p \text{ where } p \text{ is a prime}$$

Above value is always integer. Refer the proofs. The key observation is that the product of \(n\) consecutive integers is divisible by \(n!\).

p = int(1e9)+7 #example. this is a prime number
def anmodp(a, n): # a^n mod p 
    if n==1:
        return a%p

    tmp = anmodp(a, n/2)
    if n%2==1: #odd
        return (a*(tmp**2))%p
    else:
        return (tmp**2)%p 

def fac(n): # n! mod p 
    k = 1
    for i in range(2, n+1):
        k *= i
        k %= p
    return k

def invfac(n): # 1/(n!) (mod p)
    return anmodp(fac(n), p-2) #euler totient

def comb(x): # input = int array
    s = sum(x)
    ans = fac(s)
    for i in x:
        ans *= invfac(i)
    return ans % p