Find the number of Zeros -
`100!xx200!`

Options

A. 49
B. 24
C. 73
D. None of these

Answer

To find the number of trailing zeros in the expression \(100! + 200!\), we need to determine the number of trailing zeros in each factorial separately and then take the minimum of the two results.

### Step-by-Step Solution:

1. **Count the number of trailing zeros in \(100!\)**:
- To find the number of trailing zeros in a factorial, we use the formula:
\[ \text{Number of zeros} = \left\lfloor \frac{n}{5} \right\rfloor + \left\lfloor \frac{n}{5^2} \right\rfloor + \left\lfloor \frac{n}{5^3} \right\rfloor + \ldots \]
- For \(n = 100\):
- Calculate \(\left\lfloor \frac{100}{5} \right\rfloor = 20\)
- Calculate \(\left\lfloor \frac{100}{25} \right\rfloor = 4\)
- Calculate \(\left\lfloor \frac{100}{125} \right\rfloor = 0\) (since \(125 > 100\))
- Total trailing zeros in \(100!\):
\[ 20 + 4 + 0 = 24 \]

2. **Count the number of trailing zeros in \(200!\)**:
- For \(n = 200\):
- Calculate \(\left\lfloor \frac{200}{5} \right\rfloor = 40\)
- Calculate \(\left\lfloor \frac{200}{25} \right\rfloor = 8\)
- Calculate \(\left\lfloor \frac{200}{125} \right\rfloor = 1\)
- Total trailing zeros in \(200!\):
\[ 40 + 8 + 1 = 49 \]

3. **Determine the minimum number of trailing zeros**:
- Now we have:
- Trailing zeros in \(100! = 24\)
- Trailing zeros in \(200! = 49\)
- The required answer is the minimum of the two:
\[ \min(24, 49) = 24 \]

### Final Answer:
The number of trailing zeros in \(100! + 200!\) is **24**.