#Python 3, <s>93</s> 91
Thanks to [Dennis][1] and [FryAmTheEggman][2].
<!-- language-all: lang-python -->

    f=lambda x:0**x or x*f(x-1)
    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 #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)


  [1]: https://codegolf.stackexchange.com/users/12012/dennis
  [2]: https://codegolf.stackexchange.com/users/31625/fryamtheeggman