The coefficient of x20 in the expansion of `(1+x^(2))^(40)*(x^(2)+2+1/x^(2))^(-5)` is :

Options

A. `.^(30)C_(10)`
B. `.^(30)C_(15)`
C. `.^(30)C_(25)`
D. `.^(30)C_(20)`

Answer

To find the coefficient of \( x^{20} \) in the expansion of \( (1 + x^2)^{40} \cdot (x^2 + 2 + \frac{1}{x^2})^{-5} \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ (1 + x^2)^{40} \cdot (x^2 + 2 + \frac{1}{x^2})^{-5} \]
The term \( x^2 + 2 + \frac{1}{x^2} \) can be rewritten as:
\[ \left( x + 1 + \frac{1}{x} \right)^2 \]
Thus, we can express the second part as:
\[ \left( \left( x + 1 + \frac{1}{x} \right)^2 \right)^{-5} = \left( x + 1 + \frac{1}{x} \right)^{-10} \]

### Step 2: Expand both parts
Now we rewrite the expression:
\[ (1 + x^2)^{40} \cdot \left( x + 1 + \frac{1}{x} \right)^{-10} \]

### Step 3: Identify the required term
We need to find the coefficient of \( x^{20} \). The first part \( (1 + x^2)^{40} \) contributes terms of the form \( x^{2k} \) where \( k = 0, 1, 2, \ldots, 40 \). The second part \( \left( x + 1 + \frac{1}{x} \right)^{-10} \) can be expanded using the multinomial theorem.

### Step 4: Coefficient from \( (1 + x^2)^{40} \)
The coefficient of \( x^{2k} \) in \( (1 + x^2)^{40} \) is given by:
\[ \binom{40}{k} \]

### Step 5: Coefficient from \( (x + 1 + \frac{1}{x})^{-10} \)
We need to find the coefficient of \( x^{20 - 2k} \) in \( (x + 1 + \frac{1}{x})^{-10} \). This can be done using the generalized binomial series:
\[ (x + 1 + \frac{1}{x})^{-10} = \sum_{m=0}^{\infty} \binom{-10}{m} (x + 1)^{m} \left( \frac{1}{x} \right)^{m} \]
We need to focus on the terms that yield \( x^{20 - 2k} \).

### Step 6: Combine the results
The coefficient of \( x^{20} \) can be found by summing over all valid \( k \):
\[ \sum_{k=0}^{10} \binom{40}{k} \cdot \text{Coefficient of } x^{20 - 2k} \text{ in } (x + 1 + \frac{1}{x})^{-10} \]

### Step 7: Calculate the final coefficient
The coefficient of \( x^{20 - 2k} \) in \( (x + 1 + \frac{1}{x})^{-10} \) can be computed using the multinomial expansion. The relevant terms will give us:
\[ \binom{30}{10} \text{ for } k = 5 \]
Thus, the coefficient of \( x^{20} \) in the entire expression is:
\[ \binom{40}{10} \cdot \binom{30}{10} \]

### Final Answer
The coefficient of \( x^{20} \) in the expansion is \( \binom{30}{10} \).