Find the no. of zeros in expression `10 xx 100xx1000xx10000xx...10000000000` is :

Options

A. 10
B. 100
C. 50
D. 55

Explanation

To find the number of zeros in the expression \( 10 \times 100 \times 1000 \times 10000 \times \ldots \times 10000000000 \), we can start by rewriting each term in the product in terms of powers of 10.

1. **Rewrite each term**:
- \( 10 = 10^1 \)
- \( 100 = 10^2 \)
- \( 1000 = 10^3 \)
- \( 10000 = 10^4 \)
- \( 100000 = 10^5 \)
- \( 1000000 = 10^6 \)
- \( 10000000 = 10^7 \)
- \( 100000000 = 10^8 \)
- \( 1000000000 = 10^9 \)
- \( 10000000000 = 10^{10} \)

2. **Express the entire product**:
The expression can now be rewritten as:
\[ 10^1 \times 10^2 \times 10^3 \times 10^4 \times 10^5 \times 10^6 \times 10^7 \times 10^8 \times 10^9 \times 10^{10} \]

3. **Combine the powers of 10**:
When multiplying powers with the same base, we add the exponents:
\[ 10^{1+2+3+4+5+6+7+8+9+10} \]

4. **Calculate the sum of the exponents**:
The sum \( 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 \) can be calculated using the formula for the sum of the first \( n \) natural numbers:
\[ \text{Sum} = \frac{n(n + 1)}{2} \]
Here, \( n = 10 \):
\[ \text{Sum} = \frac{10 \times 11}{2} = 55 \]

5. **Final expression**:
Thus, the expression simplifies to:
\[ 10^{55} \]

6. **Determine the number of zeros**:
The number of zeros in \( 10^{55} \) is simply the exponent, which is 55.

So, the number of zeros in the expression \( 10 \times 100 \times 1000 \times 10000 \times \ldots \times 10000000000 \) is **55**.