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

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

we can follow these steps:

### Step 1: Determine the Maximum Power of \(x\)

The maximum power of \(x\) in \(f(x)\) can be calculated by summing the powers of \(x\) from each factor:

\[ 1 + 2 + 3 + \ldots + 11. \]

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

\[ S_n = \frac{n(n+1)}{2}, \]

we find:

\[ S_{11} = \frac{11 \times 12}{2} = 66. \]

Thus, the maximum power of \(x\) in \(f(x)\) is \(66\).

### Step 2: Finding the Coefficient of \(x^{60}\)

To find the coefficient of \(x^{60}\), we need to consider how we can form \(x^{60}\) from \(x^{66}\) by selecting terms that will reduce the total degree by \(6\) (since \(66 - 60 = 6\)).

### Step 3: Identify Combinations to Reduce Power

We can reduce the power of \(x\) by choosing constant terms from the factors. The goal is to select terms such that the total degree we lose sums to \(6\).

We can select from the following cases:

1. **Case 1**: Choose \(1\) from \(x-1\) and \(-6\) from \(x^6-6\).
2. **Case 2**: Choose \(-2\) from \(x^2-2\) and \(-4\) from \(x^4-4\).
3. **Case 3**: Choose \(-1\) from \(x-1\) and \(-5\) from \(x^5-5\).
4. **Case 4**: Choose \(-1\) from \(x-1\), \(-2\) from \(x^2-2\), and \(-3\) from \(x^3-3\).

### Step 4: Calculate Contributions from Each Case

1. **From Case 1**: Contribution = \(-6\).
2. **From Case 2**: Contribution = \(-2 \times -4 = 8\).
3. **From Case 3**: Contribution = \(-1 \times -5 = 5\).
4. **From Case 4**: Contribution = \(-1 \times -2 \times -3 = -6\).

### Step 5: Sum All Contributions

Now, we sum all contributions:

\[ -6 + 8 + 5 - 6 = 1. \]

### Final Answer

The coefficient of \(x^{60}\) in the polynomial function \(f(x)\) is

\[ \boxed{1}. \]