The coefficient of `x^9` in the expansion of `(1+x)(1+ x^2)(1+x^3)....(1+x^(100))` is

Explanation

To find the coefficient of \( x^9 \) in the expansion of \( (1+x)(1+x^2)(1+x^3)\ldots(1+x^{100}) \), we can approach the problem by determining how many different combinations of terms from the factors can yield \( x^9 \).

### Step-by-Step Solution:

1. **Understanding the Expansion**:
The expression \( (1+x)(1+x^2)(1+x^3)\ldots(1+x^{100}) \) can be expanded by choosing either \( 1 \) or \( x^k \) from each factor \( (1+x^k) \). The goal is to find combinations of \( k \) such that the sum of the chosen \( k \) values equals \( 9 \).

2. **Finding Combinations**:
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