Highest power of given primen number in all binomial coefficients of n

Question

Given prime number p and integer n, for what value of i where 0$\leq$i$\leq$n, the highest of p in ${n \choose i}$ occurs.

Answer 1

Kummer's theorem on binomial coefficients tells you that the exponent of $p$ occurring in ${n \choose i}$ is the number of carries when adding $n-i$ to $i$ in base $p$. So you want to choose $i$ to maximize this number of carries. The maximum number of carries will occur with $i = p^m-1$ where $p^m \le n < p^{m+1}$ (this generally won't be the only solution).