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*.