Added the justification via counting the number of permutations.

]]>I added

In combinatorics, the definition usually extends to $k = 0$ by setting $0! = 1$.

The same for binomial/multinomial coefficients, the numbers $k_i$ and $n$ can be zero as it is in Pascal triangle, then one uses $0! = 1$. (In fact, there are also the standard conventions when some of the numbers are out of bound when the multinomial coefficients are, in combinatorics, taken to be zero.) Our entry binomial theorem anyway uses the summation where the lower index is allowed to be zero as it should be.

]]>I had occasion to create minimal entries for basic combinatorial concepts: *factorial*, *multinomial coefficient*