In the polynomial `(x-1)(x^(2)-2)(x^(3)-3)…(x^(11)-11)`, the coefficient of `x^(60)` is :

Answer

To find the coefficient of \( x^{60} \) in the polynomial \( (x-1)(x^2-2)(x^3-3)\ldots(x^{11}-11) \), we can follow these steps:

### Step 1: Determine the total degree of the polynomial
The polynomial is formed by multiplying terms of the form \( (x^k - k) \) for \( k = 1 \) to \( 11 \). The total degree of the polynomial is the sum of the degrees of each term:
\[ 1 + 2 + 3 + \ldots + 11 = \frac{11 \times 12}{2} = 66 \]
This means the highest degree term in the polynomial is \( x^{66} \).

**Hint:** Use the formula for the sum of the first \( n \) natural numbers to find the total degree.

### Step 2: Relate the desired degree to the total degree
We want the coefficient of \( x^{60} \). Since the total degree is \( 66 \), we can express \( x^{66} \) as:
\[ x^{66} = x^{60} \cdot x^6 \]
This indicates that we need to account for \( x^6 \) in our polynomial.

**Hint:** Think about how you can form \( x^{60} \) by reducing the total degree from \( 66 \) to \( 60 \).

### Step 3: Identify how to obtain \( x^6 \)
To obtain \( x^6 \) from the polynomial, we can choose to omit \( x^6 - 6 \) from the product and instead take the constant term \( -6 \) from this term. This means we will include all other terms \( (x^k - k) \) for \( k \neq 6 \).

**Hint:** Consider what happens when you exclude a term from a product and how that affects the overall polynomial.

### Step 4: Calculate the contribution of the omitted term
By omitting \( (x^6 - 6) \), we are left with the product:
\[ (x-1)(x^2-2)(x^3-3)(x^4-4)(x^5-5)(x^7-7)(x^8-8)(x^9-9)(x^{10}-10)(x^{11}-11) \]
The coefficient of \( x^{60} \) in this modified polynomial will be the same as the coefficient of \( x^{60} \) in the original polynomial, but now we have to add the contribution from \( -6 \).

**Hint:** Think about how the coefficients of the remaining terms contribute to the overall polynomial.

### Step 5: Find the coefficient of \( x^{60} \)
The coefficient of \( x^{60} \) in the original polynomial, after considering the contribution from \( -6 \), is:
\[ \text{Coefficient of } x^{60} = -6 \]

### Final Answer
Thus, the coefficient of \( x^{60} \) in the polynomial \( (x-1)(x^2-2)(x^3-3)\ldots(x^{11}-11) \) is:
\[ \boxed{-6} \]