For the proof we can allow of numbers in base $p$ with leading zeros. So let

all in base $p$. We set $r=n-m$ and denote the $p$-adic representation of $r$ with $r_{d}r_{{d-1}}\cdots r_{0}$.

We define $c_{{-1}}=0$, and for each $0\leq j\leq d$

Finally, we introduce $\delta_{p}(n)$ as the sum of digits in the $p$-adic of $n$. Then it follows that the power of $p$ dividing $\binom{n}{m}$ is

For each $j\geq 0$, we have

Then

This gives us

the total number of carries. ∎