In the expansion of `(x-1) (x-2) …(x-18)`, the coefficient of `x^(17)` is

Options

A. `684`
B. `-171`
C. `171`
D. `-342`

Answer

To find the coefficient of \( x^{17} \) in the expansion of \( (x-1)(x-2)(x-3)\ldots(x-18) \), we can follow these steps:

### Step 1: Understand the Expansion
The expression \( (x-1)(x-2)(x-3)\ldots(x-18) \) is a polynomial of degree 18. The coefficient of \( x^{17} \) corresponds to the sum of the products of the roots taken one at a time (with a negative sign, since each term contributes a negative root).

### Step 2: Identify the Roots
The roots of the polynomial are \( 1, 2, 3, \ldots, 18 \).

### Step 3: Calculate the Coefficient of \( x^{17} \)
The coefficient of \( x^{17} \) can be found by taking the negative sum of the roots:
\[ \text{Coefficient of } x^{17} = -(1 + 2 + 3 + \ldots + 18) \]

### Step 4: Use the Formula for the Sum of the First \( n \) Natural Numbers
The sum of the first \( n \) natural numbers is given by the formula:
\[ S_n = \frac{n(n+1)}{2} \]
For \( n = 18 \):
\[ S_{18} = \frac{18 \times 19}{2} = \frac{342}{2} = 171 \]

### Step 5: Apply the Result
Thus, the coefficient of \( x^{17} \) is:
\[ -(1 + 2 + 3 + \ldots + 18) = -171 \]

### Final Answer
The coefficient of \( x^{17} \) in the expansion of \( (x-1)(x-2)(x-3)\ldots(x-18) \) is \( -171 \).

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