#Python 3, 93

<!-- language-all: lang-python -->

    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