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](https://mathoverflow.net/questions/80169/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*.



p-adic valuation for multinomial coefficients
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
$$\operatorname{ord}\_p \binom{n}{k\_1,\ldots,k\_r}=\frac{\sum\_{i=1}^rS(k\_i)-S(n)}{p-1}.$$


