The term independent of x in the expansion of `((1)/(x^(2)) + (1)/(x) +1 + x + x^(2))^(5)`, is

Options

A. 381
B. 441
C. 439
D. 359

Explanation

To find the term independent of \( x \) in the expansion of

\[ \left( \frac{1}{x^2} + \frac{1}{x} + 1 + x + x^2 \right)^5, \]

we can follow these steps:

### Step 1: Rewrite the expression
We start by rewriting the expression in a more manageable form:

\[ \left( \frac{1}{x^2} + \frac{1}{x} + 1 + x + x^2 \right)^5. \]

### Step 2: Identify the powers of \( x \)
We can express each term in the expansion in terms of powers of \( x \):

- \( \frac{1}{x^2} \) corresponds to \( x^{-2} \)
- \( \frac{1}{x} \) corresponds to \( x^{-1} \)
- \( 1 \) corresponds to \( x^0 \)
- \( x \) corresponds to \( x^1 \)
- \( x^2 \) corresponds to \( x^2 \)

### Step 3: Use the multinomial theorem
We can use the multinomial theorem to expand the expression. The general term in the expansion will be of the form:

\[ \frac{5!}{a_1! a_2! a_3! a_4! a_5!} \left( \frac{1}{x^2} \right)^{a_1} \left( \frac{1}{x} \right)^{a_2} (1)^{a_3} (x)^{a_4} (x^2)^{a_5}, \]

where \( a_1 + a_2 + a_3 + a_4 + a_5 = 5 \).

### Step 4: Find the exponent of \( x \)
The exponent of \( x \) in the general term is given by:

\[ -2a_1 - a_2 + a_4 + 2a_5. \]

We want to find the values of \( a_1, a_2, a_3, a_4, a_5 \) such that the exponent of \( x \) is zero:

\[ -2a_1 - a_2 + a_4 + 2a_5 = 0. \]

### Step 5: Solve for combinations
We also have the constraint:

\[ a_1 + a_2 + a_3 + a_4 + a_5 = 5. \]

We can express \( a_3 \) in terms of the other variables:

\[ a_3 = 5 - (a_1 + a_2 + a_4 + a_5). \]

### Step 6: Substitute and simplify
Now, we can substitute \( a_3 \) back into the equation and solve for different combinations of \( a_1, a_2, a_4, a_5 \) that satisfy both equations.

### Step 7: Calculate the coefficient
Once we find the valid combinations of \( a_1, a_2, a_4, a_5 \), we can calculate the coefficient of the term using the multinomial coefficient.

### Step 8: Find the independent term
Finally, we sum the coefficients of all the terms that are independent of \( x \).