2 of 6 added 20 characters in body

Python 3, 109 103

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

n as integer, k as iterable.

Ungolfed:

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