Find no of zeros in 100 !

Answer

To find the number of trailing zeros in \(100!\), we can use the method of counting the number of times 5 is a factor in the numbers from 1 to 100. This is because there are usually more factors of 2 than factors of 5 in factorials, and each pair of 2 and 5 contributes to a trailing zero.

### Step-by-Step Solution:

1. **Understand the formula**: The number of trailing zeros in \(n!\) can be calculated using the formula:
\[ \text{Number of trailing zeros} = \left\lfloor \frac{n}{5} \right\rfloor + \left\lfloor \frac{n}{5^2} \right\rfloor + \left\lfloor \frac{n}{5^3} \right\rfloor + \ldots \]
until \(n/5^k < 1\).

2. **Calculate for \(n = 100\)**:
- First, calculate \(\left\lfloor \frac{100}{5} \right\rfloor\):
\[ \left\lfloor \frac{100}{5} \right\rfloor = \left\lfloor 20 \right\rfloor = 20 \]
- Next, calculate \(\left\lfloor \frac{100}{5^2} \right\rfloor\):
\[ \left\lfloor \frac{100}{25} \right\rfloor = \left\lfloor 4 \right\rfloor = 4 \]
- Now, calculate \(\left\lfloor \frac{100}{5^3} \right\rfloor\):
\[ \left\lfloor \frac{100}{125} \right\rfloor = \left\lfloor 0.8 \right\rfloor = 0 \]

3. **Sum the values**:
- Now, we sum the results from the calculations:
\[ 20 + 4 + 0 = 24 \]

4. **Conclusion**:
- Therefore, the number of trailing zeros in \(100!\) is \(24\).

### Final Answer:
The number of trailing zeros in \(100!\) is **24**.