Coefficient of `x^((n^2+n-14)/2)` in, `(x-1)(x^2-2)(x^3-3)(x^4- 4).........(x^n-n),(n>=8)` is

Answer

To find the coefficient of \( x^{(n^2+n-14)/2} \) in the expression \( (x-1)(x^2-2)(x^3-3)(x^4-4)\ldots(x^n-n) \), we can follow these steps:

### Step 1: Determine the maximum power of \( x \)

The expression consists of terms \( (x^k - k) \) for \( k = 1, 2, \ldots, n \). The maximum power of \( x \) occurs when we take all the \( x^k \) terms:

\[ x^1 \cdot x^2 \cdot x^3 \cdots x^n = x^{1 + 2 + 3 + \ldots + n} \]

Using the formula for the sum of the first \( n \) natural numbers:

\[ 1 + 2 + 3 + \ldots + n = \frac{n(n+1)}{2} \]

Thus, the maximum power of \( x \) is:

\[ x^{\frac{n(n+1)}{2}} \]

### Step 2: Identify the target power of \( x \)

We need to find the coefficient of \( x^{(n^2+n-14)/2} \). This can be rewritten as:

\[ x^{\frac{n(n+1)}{2} - 7} \]

This indicates that we need to remove \( 7 \) from the total power of \( x \).

### Step 3: Determine how to achieve the removal of \( 7 \)

To remove \( 7 \) from the power, we can select terms from the product \( (x^k - k) \) such that the sum of the constants we choose equals \( 7 \). The constants are \( 1, 2, 3, \ldots, n \).

### Step 4: Find combinations of constants that sum to \( 7 \)

We need to find combinations of the integers \( 1, 2, \ldots, n \) that sum to \( 7 \). The possible combinations are:

1. \( 7 \)
2. \( 6 + 1 \)
3. \( 5 + 2 \)
4. \( 4 + 3 \)
5. \( 4 + 2 + 1 \)
6. \( 3 + 2 + 2 \) (not valid since we can only use each number once)
7. \( 3 + 4 \) (already counted)
8. \( 2 + 5 \) (already counted)
9. \( 1 + 6 \) (already counted)

The valid combinations that sum to \( 7 \) are:

- \( 7 \)
- \( 6 + 1 \)
- \( 5 + 2 \)
- \( 4 + 3 \)
- \( 4 + 2 + 1 \)

### Step 5: Calculate the contribution of each combination

1. **Using \( 7 \)**: Coefficient is \( -1 \).
2. **Using \( 6 + 1 \)**: Coefficient is \( -6 \cdot -1 = 6 \).
3. **Using \( 5 + 2 \)**: Coefficient is \( -5 \cdot -2 = 10 \).
4. **Using \( 4 + 3 \)**: Coefficient is \( -4 \cdot -3 = 12 \).
5. **Using \( 4 + 2 + 1 \)**: Coefficient is \( -4 \cdot -2 \cdot -1 = -8 \).

### Step 6: Sum the contributions

Now we sum the contributions from all valid combinations:

\[ -1 + 6 + 10 + 12 - 8 = 19 \]

### Final Result

Thus, the coefficient of \( x^{(n^2+n-14)/2} \) in the given expression is:

\[ \boxed{19} \]