Find the no. of zeros in expression :
1 × 2 × 3 × 4 .......... × 500

Options

A. 124
B. 125
C. 130
D. 135

Answer

To find the number of zeros in the expression \(1 \times 2 \times 3 \times \ldots \times 500\) (which is \(500!\)), we need to determine how many times \(10\) is a factor in this product. Since \(10\) is made up of \(2\) and \(5\), we need to find the number of pairs of these factors. However, there are usually more factors of \(2\) than \(5\) in factorials, so we will focus on counting the number of \(5\)s.

### Step-by-Step Solution:

1. **Count the multiples of \(5\)**:
We first count how many numbers from \(1\) to \(500\) are multiples of \(5\).
\[ \text{Number of multiples of } 5 = \left\lfloor \frac{500}{5} \right\rfloor = 100 \]

2. **Count the multiples of \(25\)**:
Next, we count how many numbers are multiples of \(25\) (since \(25 = 5^2\) contributes an additional \(5\)).
\[ \text{Number of multiples of } 25 = \left\lfloor \frac{500}{25} \right\rfloor = 20 \]

3. **Count the multiples of \(125\)**:
We also count the multiples of \(125\) (since \(125 = 5^3\) contributes yet another \(5\)).
\[ \text{Number of multiples of } 125 = \left\lfloor \frac{500}{125} \right\rfloor = 4 \]

4. **Count the multiples of \(625\)**:
Finally, we check for multiples of \(625\) (since \(625 = 5^4\)), but since \(625 > 500\), this count is \(0\).
\[ \text{Number of multiples of } 625 = \left\lfloor \frac{500}{625} \right\rfloor = 0 \]

5. **Total count of factors of \(5\)**:
Now, we sum all these counts to find the total number of factors of \(5\) in \(500!\):
\[ \text{Total factors of } 5 = 100 + 20 + 4 + 0 = 124 \]

Thus, the number of trailing zeros in the expression \(1 \times 2 \times 3 \times \ldots \times 500\) is **124**.

### Final Answer:
The number of zeros in the expression \(1 \times 2 \times 3 \times \ldots \times 500\) is **124**.