TITLE: p-adic valuation for multinomial coefficients
QUESTION [9 upvotes]: Kummer's formula
https://en.wikipedia.org/w/index.php?title=Kummer%27s_theorem&oldid=745783657
says that
$$
  \text{ord}_p \binom{n}{k}
$$
is the number of carries required when adding the base-$p$ expansions
of $k$ and $n-k$. Is there a similar formula for the $p$-adic
valuation of a multinomial coefficient
$$
  \binom{n}{k_1,\ldots,k_r} := \frac{n!}{k_1!\cdots k_r!} ?
$$
If so, is there a good reference (free online for preference, but failing that,
in a book)?
There is a related question
Reference needed for Lucas' Theorem for multinomial coefficients modulo a prime,
but it involves the value of the multinomial coefficient modulo $p$, not the $p$-adic valuation.

REPLY [15 votes]: Denote the sum of the digits of $n$ in base $b$ by $S(n)$. Then the number of carries when adding $k_1+k_2$ is 
$$\frac{1}{b-1}\big(S(k_1)+S(k_2)-S(k_1+k_2)\big).$$
This shows that the number of carries when successively adding $(((k_1+k_2)+k_3)+\cdots +k_r)$ is
$$\frac{1}{b-1}\left(\sum_{i=1}^r S(k_i)-S\left(\sum_{i=1}^r k_i\right)\right),$$
and this last expression is clearly independent of the order in which they are added. The formula for the multinomial coefficient can thus be written as
$$\operatorname{ord}_p \binom{n}{k_1,\ldots,k_r}=\frac{\sum_{i=1}^rS(k_i)-S(n)}{p-1}.$$