Firstly, 2 zeros are produced by the multiplication of 100.
Also there will be another 8 zeros from the multiplication of the numbers: 10, 20, 30, 40, 60, 70, 80, 90.
But, 25, 50 and 75 will give 2 fives each, since 25= 5*5, 50= 5*5*2 and 75= 5*5*3.
This 5s will be multiplied by another 6 even numbers to give another 6 zeros.
Then, there are 8 other numbers between 1 and 100 that have 5 in the units.
( 5, 15, 35, 45, 55, 65, 85, 95).
These 8 numbers will be multiplied by any 8 even numbers to produce another 8 zeros.
All in all, in the end of 100! will be:
8+6+8+2= 24 zeros.