Find the number of Zeros -
100! +200!

Options

A. 24
B. 25
C. 49
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 both \(100!\) and \(200!\) separately, and then take the minimum of the two results. The number of trailing zeros in a factorial can be found by counting the factors of 5 in the numbers from 1 to that factorial number.

### Step-by-step Solution:

1. **Calculate the number of trailing zeros in \(100!\)**:
- We use the formula to find the number of trailing zeros in \(n!\):
\[ \text{Number of trailing zeros} = \left\lfloor \frac{n}{5} \right\rfloor + \left\lfloor \frac{n}{25} \right\rfloor + \left\lfloor \frac{n}{125} \right\rfloor + \ldots \]
- For \(n = 100\):
\[ \left\lfloor \frac{100}{5} \right\rfloor = 20 \]
\[ \left\lfloor \frac{100}{25} \right\rfloor = 4 \]
\[ \left\lfloor \frac{100}{125} \right\rfloor = 0 \]
- Adding these values together:
\[ \text{Total zeros in } 100! = 20 + 4 + 0 = 24 \]

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

3. **Determine the number of trailing zeros in \(100! + 200!\)**:
- Since \(100!\) has 24 trailing zeros and \(200!\) has 49 trailing zeros, the number of trailing zeros in the sum \(100! + 200!\) will be determined by the smaller of the two:
- Therefore, the number of trailing zeros in \(100! + 200!\) is:
\[ \text{Minimum}(24, 49) = 24 \]

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