In `(2^(1/3)+1/(3^(1/3)))^n` if the ratio of 7th term from the beginning to the 7th term from the end is `1/6`, then the value of `n`is (a) 6 (b) 9 (c) 12 (d) 15

Options

A. `6`
B. `9`
C. `12`
D. `15`

Explanation

To solve the problem, we need to find the value of \( n \) in the expression \( (2^{1/3} + 3^{-1/3})^n \) given that the ratio of the 7th term from the beginning to the 7th term from the end is \( \frac{1}{6} \).

### Step-by-step Solution:

1. **Identify the General Term**:
The general term \( T_k \) in the binomial expansion of \( (a + b)^n \) is given by:
\[ T_k = \binom{n}{k-1} a^{n-k+1} b^{k-1} \]
Here, \( a = 2^{1/3} \) and \( b = 3^{-1/3} \).

2. **Find the 7th Term from the Beginning**:
The 7th term from the beginning is \( T_7 \):
\[ T_7 = \binom{n}{6} (2^{1/3})^{n-6} (3^{-1/3})^6 \]
Simplifying this gives:
\[ T_7 = \binom{n}{6} (2^{(n-6)/3}) (3^{-2}) \]

3. **Find the 7th Term from the End**:
The 7th term from the end is given by \( T_{n-6} \):
\[ T_{n-6} = \binom{n}{n-6} (2^{1/3})^6 (3^{-1/3})^{n-6} \]
Simplifying this gives:
\[ T_{n-6} = \binom{n}{6} (2^2) (3^{-(n-6)/3}) \]

4. **Set Up the Ratio**:
We know that:
\[ \frac{T_7}{T_{n-6}} = \frac{1}{6} \]
Substituting the expressions for \( T_7 \) and \( T_{n-6} \):
\[ \frac{\binom{n}{6} (2^{(n-6)/3}) (3^{-2})}{\binom{n}{6} (2^2) (3^{-(n-6)/3})} = \frac{1}{6} \]
Canceling \( \binom{n}{6} \):
\[ \frac{(2^{(n-6)/3}) (3^{-2})}{(2^2) (3^{-(n-6)/3})} = \frac{1}{6} \]

5. **Simplify the Equation**:
This simplifies to:
\[ \frac{2^{(n-6)/3}}{2^2} \cdot \frac{3^{(n-6)/3}}{3^2} = \frac{1}{6} \]
Which can be rewritten as:
\[ 2^{(n-6)/3 - 2} \cdot 3^{(n-6)/3 + 2} = \frac{1}{6} \]

6. **Equate the Powers**:
Since \( \frac{1}{6} = \frac{1}{2 \cdot 3} \), we can equate the powers:
\[ (n-6)/3 - 2 = -1 \quad \text{(for base 2)} \]
\[ (n-6)/3 + 2 = -1 \quad \text{(for base 3)} \]

7. **Solve for n**:
Solving the first equation:
\[ \frac{n-6}{3} = 1 \implies n - 6 = 3 \implies n = 9 \]

Solving the second equation:
\[ \frac{n-6}{3} = -3 \implies n - 6 = -9 \implies n = -3 \quad \text{(not valid)} \]

8. **Conclusion**:
The only valid solution is \( n = 9 \).

### Final Answer:
The value of \( n \) is \( 9 \) (Option b).