24 is correct for trailing zeoes of 100!...

The logic required for n!...
-only multiplying by 10 or a factor of 10 will produce a trailing zero

-multiplying 5 by an even number will produce a factor of 10

-there will be more even numbers than multiples of 5, so 5's are the limiting number in the next step.

-reduce all the numbers down to there prime numbers

-counting all the times 5 occurs will give you the number of trailing zeroes.