To find the number of zeros at the end of , we need to determine how many times divides into . Since , every pair of factors of 2 and 5 contributes a factor of 10, resulting in one zero at the end of the number.
The number of factors of 5 in will determine the number of zeros at the end, because there are always more factors of 2 than factors of 5 in any factorial. To find the number of factors of 5, we use the following formula: The first term counts the multiples of 5, and the second term counts the multiples of 25 (which contribute an extra factor of 5). Let's calculate it: Thus, the total number of factors of 5 in is: Therefore, the number of zeros at the end of is .
Answer: