Consider the expansion of `( 1+ x)^(2n+1)`
The coefficient of `x^(99)` in the expansion of `(x-1) ( x-2) (x -3)"….." (x-100)` is

Options

A. 5050
B. 5000
C. `-5050`
D. `-5000`

Answer

To find the coefficient of \( x^{99} \) in the expansion of \( (x - 1)(x - 2)(x - 3) \ldots (x - 100) \), we can follow these steps:

### Step 1: Understand the structure of the polynomial
The polynomial \( (x - 1)(x - 2)(x - 3) \ldots (x - 100) \) is a degree 100 polynomial. The general form of this polynomial can be expressed as:
\[ P(x) = x^{100} - (1 + 2 + 3 + \ldots + 100)x^{99} + \ldots \]
Here, the coefficient of \( x^{99} \) is the negative sum of the roots.

### Step 2: Calculate the sum of the roots
The roots of the polynomial are \( 1, 2, 3, \ldots, 100 \). The sum of the first \( n \) natural numbers is given by the formula:
\[ \text{Sum} = \frac{n(n + 1)}{2} \]
For \( n = 100 \):
\[ \text{Sum} = \frac{100 \times (100 + 1)}{2} = \frac{100 \times 101}{2} = 5050 \]

### Step 3: Determine the coefficient of \( x^{99} \)
From the expansion, the coefficient of \( x^{99} \) is the negative of the sum of the roots:
\[ \text{Coefficient of } x^{99} = -5050 \]

### Final Answer
Thus, the coefficient of \( x^{99} \) in the expansion of \( (x - 1)(x - 2)(x - 3) \ldots (x - 100) \) is:
\[ \boxed{-5050} \]