Number of zeros in the expansion of `100!` is

Options

A. 22
B. 23
C. 24
D. 25

Explanation

To find the number of zeros in the expansion of \(100!\), we need to determine how many times \(10\) is a factor in \(100!\). Since \(10\) can be expressed as \(2 \times 5\), we need to find the number of pairs of \(2\)s and \(5\)s in the prime factorization of \(100!\). However, since there are generally more \(2\)s than \(5\)s in factorials, the limiting factor will be the number of \(5\)s.

### Step-by-Step Solution:

1. **Identify the formula for finding the exponent of a prime \(p\) in \(n!\)**:
The exponent of a prime \(p\) in \(n!\) can be calculated using the formula:
\[
\text{Exponent of } p = \left\lfloor \frac{n}{p} \right\rfloor + \left\lfloor \frac{n}{p^2} \right\rfloor + \left\lfloor \frac{n}{p^3} \right\rfloor + \ldots
...