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*.

carry patters(=which carrys appear at which digit positions) determines the irreducible composition factors. Yours truly had the pleasure of extending and refining (to the extent it was feasible) those results to algebraic groups of types $B,C,D$. $\endgroup$ – Jyrki Lahtonen Feb 4 '19 at 19:471more comment