How many zeroes at the end of 100!?

The number of trailing zeros in the decimal representation of n!, the factorial of a non-negative integer n, can be determined with the formula:

Z(n) = floor(n/5) + floor(n/25) + floor(n/125) + ...

This formula essentially counts the number of factors of 5 in n!, because each factor of 5 will contribute one trailing zero (and factors of 2 are more common).

So, for 100!:

Z(100) = floor(100/5) + floor(100/25) + floor(100/125) + ...

= 20 + 4 + 0 + ...

= 24

So, there are 24 trailing zeros in the decimal representation of 100!.