Preimage Size of Factorial Zeroes Function

Count how many nonnegative integers x satisfy f(x)=k, where f(x) is the number of trailing zeros in x! (i.e., the sum of floor(x/5^i)). Key insight: f is nondecreasing with jumps aligned to multiples of 5, so the preimage size can be found by binary-searching the range for k (and is either 0 or 5).