The coefficient of `x^(n-2)` in the polynomial `(x-1)(x-2)(x-3)...(x-n)` is

Options

A. `(n(n^(2)+2)(3n+1))/(24)`
B. `(n(n^(2)-1)(3n+2))/(24)`
C. `(n(n^(2)+1)(3n+4))/(24)`
D. `(n(n^(2)-2)(3n-2))/(24)`

Answer

To find the coefficient of \( x^{n-2} \) in the polynomial \( (x-1)(x-2)(x-3)\cdots(x-n) \), we can follow these steps:

### Step 1: Understand the Polynomial
The polynomial \( (x-1)(x-2)(x-3)\cdots(x-n) \) is a degree \( n \) polynomial. The general form of this polynomial can be expressed as:
\[ P(x) = x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + \ldots + a_0 \]
where \( a_{n-2} \) is the coefficient we want to find.

### Step 2: Use Vieta's Formulas
According to Vieta's formulas, the coefficients of the polynomial can be expressed in terms of the roots. The roots of the polynomial are \( 1, 2, 3, \ldots, n \).

### Step 3: Coefficient of \( x^{n-2} \)
The coefficient of \( x^{n-2} \) is given by the sum of the products of the roots taken \( n-2 \) at a time, with a negative sign:
\[ a_{n-2} = -\sum_{1 \leq i < j \leq n} ij \]
This means we need to find the sum of the products of the roots taken two at a time.

### Step 4: Calculate the Required Sum
We can calculate the sum of the products of the roots \( ij \) for \( i \) and \( j \) ranging from \( 1 \) to \( n \):
\[ \sum_{1 \leq i < j \leq n} ij = \frac{1}{2} \left( \left( \sum_{k=1}^{n} k \right)^2 - \sum_{k=1}^{n} k^2 \right) \]
where \( \sum_{k=1}^{n} k = \frac{n(n+1)}{2} \) and \( \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6} \).

### Step 5: Substitute the Sums
Substituting these sums into our equation:
\[ \sum_{1 \leq i < j \leq n} ij = \frac{1}{2} \left( \left( \frac{n(n+1)}{2} \right)^2 - \frac{n(n+1)(2n+1)}{6} \right) \]

### Step 6: Simplify the Expression
1. Calculate \( \left( \frac{n(n+1)}{2} \right)^2 \):
\[ = \frac{n^2(n+1)^2}{4} \]
2. Calculate \( \frac{n(n+1)(2n+1)}{6} \).
3. Combine these results to find the coefficient \( a_{n-2} \).

### Step 7: Final Coefficient
After simplification, the coefficient \( a_{n-2} \) can be expressed as:
\[ a_{n-2} = -\frac{n(n+1)(n-1)}{6} \]

Thus, the coefficient of \( x^{n-2} \) in the polynomial \( (x-1)(x-2)(x-3)\cdots(x-n) \) is:
\[ -\frac{n(n+1)(n-1)}{6} \]