In the polynomial function `f(x)=(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 function

\[ f(x) = (x-1)(x^2-2)(x^3-3) \ldots (x^{11}-11), \]

we will follow these steps:

### Step 1: Determine the total degree of the polynomial
First, we need to find the total degree of the polynomial \( f(x) \). The degrees of the individual factors are:

- \( x-1 \) contributes degree 1,
- \( x^2-2 \) contributes degree 2,
- \( x^3-3 \) contributes degree 3,
- ...
- \( x^{11}-11 \) contributes degree 11.

To find the total degree, we sum these degrees:

\[ 1 + 2 + 3 + \ldots + 11 = \frac{11 \times (11 + 1)}{2} = \frac{11 \times 12}{2} = 66. \]

So, the total degree of \( f(x) \) is 66.

### Step 2: Express \( x^{60} \) in terms of the total degree
We want to find the coefficient of \( x^{60} \). Since the total degree is 66, we can express \( x^{60} \) as:

\[ x^{60} = x^{66} \cdot x^{-6}. \]

This means we need to exclude terms that contribute to \( x^6 \) from the expansion of \( f(x) \).

### Step 3: Identify terms to exclude
To obtain \( x^{60} \), we need to exclude one of the terms that contribute to the total degree of 66. The term we will exclude must be such that its degree sums to 6. The possible terms that can be excluded are:

- Exclude \( -1 \) from \( (x-1) \) (degree 1),
- Exclude \( -2 \) from \( (x^2-2) \) (degree 2),
- Exclude \( -3 \) from \( (x^3-3) \) (degree 3),
- Exclude \( -4 \) from \( (x^4-4) \) (degree 4),
- Exclude \( -5 \) from \( (x^5-5) \) (degree 5),
- Exclude \( -6 \) from \( (x^6-6) \) (degree 6).

### Step 4: Calculate the coefficient of \( x^{60} \)
To find the coefficient of \( x^{60} \), we will consider the contributions from excluding each of these terms:

1. Excluding \( -1 \) contributes \( -1 \) (from \( (x-1) \)).
2. Excluding \( -2 \) contributes \( -2 \) (from \( (x^2-2) \)).
3. Excluding \( -3 \) contributes \( -3 \) (from \( (x^3-3) \)).
4. Excluding \( -4 \) contributes \( -4 \) (from \( (x^4-4) \)).
5. Excluding \( -5 \) contributes \( -5 \) (from \( (x^5-5) \)).
6. Excluding \( -6 \) contributes \( -6 \) (from \( (x^6-6) \)).

The total contribution to the coefficient of \( x^{60} \) is:

\[ -1 - 2 - 3 - 4 - 5 - 6 = -21. \]

### Final Result
Thus, the coefficient of \( x^{60} \) in \( f(x) \) is

\[ \boxed{-21}. \]