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 $$\operatorname{ord}_p \binom{n}{k_1,\ldots,k_r}=\frac{\sum_{i=1}^rS(k_i)-S(n)}{p-1}.$$

You can write:

$$\binom{n}{k_1,\ldots,k_r} = \frac{n!}{k_1!\cdots k_r!}$$

as:

$$\binom{n}{k_1,\ldots,k_r} = \binom{n}{k_1}\binom{n-k_1}{k_2}\ldots\binom{n-k_1\ldots-k_{r-1}}{k_r}$$