# How many zero's are there in 100! (100 factorial) ? Plz explain and answer.

##### 1 Answer
Mar 28, 2018

The number of zeros in 100! will be $24$.

#### Explanation:

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 $24$.