The Combinatorics of Kummer's Theorem on Binomial Coefficients

Source abstract

We present novel combinatorial proofs for some cases of a classic result of Kummer, that the highest power of $p$ dividing the binomial coefficient $n$ choose $m$ equals the number of carries when adding $m$ to $n-m$ in base $p$. We prove the case $p=2$ by considering permutations of binary vectors, and extend this approach to arbitrary primes for the case of no carries.