The Combinatorics of Kummer's Theorem on Binomial Coefficients

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.