I understand number of zeros means number of zeros at the end of 100! i.e. trailing zeros.
If you dot know, 100! =100xx99xx98xx… xx2xx1
How are the trailing zeros are formed. A trailing zero will be formed when a multiple of 5 is multiplied with a multiple of 2. How many do we have in this long product?
First we should count the 5’s - 5,10,15,20,25 and so on i.e. a total of 20. However 25,50,75 and 100 have two 5’s so for each of them, you count them twice, which makes for total 24.
Now to count the number of 2’s - 2,4,6,8,10 and so on i.e. a total of 50 multiples of 2’s, 25 multiples of 4’s (giving us 25 more 2's), 12 multiples of 8’s (giving us 12 more 2's) and so on… i.e. far more than 24
Now as each pair of 2 and 5 will give a trailing zero, but we have only 24 5’s and far more 2's,
we can only make 24 such pairs and
hence, the number of zeros in 100! will be 24.