The sum of the binomial coefficients of the expansion `(x+(1)/(x))^(n)` is equal to 256. The term independent of x is

Options

A. `T_(3)`
B. `T_(4)`
C. `T_(5)`
D. none

Answer

To solve the problem, we need to find the term independent of \( x \) in the expansion of \( \left( x + \frac{1}{x} \right)^n \) given that the sum of the binomial coefficients is equal to 256.

### Step-by-Step Solution:

1. **Understanding the Binomial Expansion**:
The binomial expansion of \( \left( x + \frac{1}{x} \right)^n \) can be expressed using the binomial theorem:
\[ T_{r+1} = \binom{n}{r} x^{n-r} \left( \frac{1}{x} \right)^r = \binom{n}{r} x^{n-2r} \]
Here, \( T_{r+1} \) is the \( (r+1)^{th} \) term in the expansion.

2. **Finding the Sum of Binomial Coefficients**:
The sum of the binomial coefficients of the expansion is given by:
\[ \sum_{r=0}^{n} \binom{n}{r} = 2^n \]
We know from the problem that this sum equals 256. Therefore:
\[ 2^n = 256 \]
Since \( 256 = 2^8 \), we have:
\[ n = 8 \]

3. **Finding the Term Independent of \( x \)**:
We need to find the term where the power of \( x \) is zero, i.e., \( n - 2r = 0 \):
\[ 8 - 2r = 0 \implies 2r = 8 \implies r = 4 \]
Thus, the term independent of \( x \) corresponds to \( r = 4 \).

4. **Calculating the Independent Term**:
Now, we substitute \( r = 4 \) into the general term:
\[ T_{4+1} = T_5 = \binom{8}{4} x^{8 - 2 \cdot 4} = \binom{8}{4} x^0 = \binom{8}{4} \]
We need to calculate \( \binom{8}{4} \):
\[ \binom{8}{4} = \frac{8!}{4! \cdot 4!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70 \]

5. **Final Answer**:
The term independent of \( x \) is:
\[ \boxed{70} \]