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Python 3, 93

from math import*
f=factorial
def g(n,k):
    r=f(n)
    for i in k:r//=f(i)
    return r//f(n-sum(k))

n as integer, k as iterable.

Ungolfed:

import functools #reduce
import math #factorial
import operator #mul
def multinomial(n, k):
    km = n - sum(k)
    k.append(km)
    numerator = math.factorial(n)
    denominator = functools.reduce(operator.mul, (math.factorial(i) for i in k)) #product of iterable k
    return numerator//denominator