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Python 3, 93 91

Thanks to Dennis and FryAmTheEggman.

f=lambda x:0**x or x*f(x-1)
def g(n,k):
    for i in k:r//=f(i)
    return r//f(n-sum(k))

n as integer, k as iterable.


import functools #cache

@functools.lru_cache(maxsize=None) #cache results to speed up calculations
def factorial(x):
    if x <= 1: return 1
    else: return x * factorial(x-1)

def multinomial(n, k):
    ret = factorial(n)
    for i in k: ret //= factorial(i)
    km = n - sum(k)
    return ret//factorial(km)