If the coefficients of `x^7 and x^8` in the expansion of `[2 +x/3]^n` are equal, then the value of n is : (A) 15 (B) 45 (C) 55 (D) 56

Options

A. 15
B. 45
C. 55
D. 60

Explanation

To solve the problem, we need to find the value of \( n \) such that the coefficients of \( x^7 \) and \( x^8 \) in the expansion of \( \left(2 + \frac{x}{3}\right)^n \) are equal.

### Step-by-Step Solution:

1. **Identify the General Term**:
The general term in the binomial expansion of \( \left(2 + \frac{x}{3}\right)^n \) is given by:
\[ T_{r+1} = \binom{n}{r} \cdot 2^{n-r} \cdot \left(\frac{x}{3}\right)^r \]
This simplifies to:
\[ T_{r+1} = \binom{n}{r} \cdot 2^{n-r} \cdot \frac{x^r}{3^r} \]

2. **Find the Coefficient of \( x^7 \)**:
For \( x^7 \), we set \( r = 7 \):
\[ T_{8} = \binom{n}{7} \cdot 2^{n-7} \cdot \frac{x^7}{3^7} \]
The coefficient of \( x^7 \) is:
\[ C_7 = \binom{n}{7} \cdot 2^{n-7} \cdot \frac{1}{3^7} \]

3. **Find the Coefficient of \( x^8 \)**:
For \( x^8 \), we set \( r = 8 \):
\[ T_{9} = \binom{n}{8} \cdot 2^{n-8} \cdot \frac{x^8}{3^8} \]
The coefficient of \( x^8 \) is:
\[ C_8 = \binom{n}{8} \cdot 2^{n-8} \cdot \frac{1}{3^8} \]

4. **Set the Coefficients Equal**:
According to the problem, the coefficients of \( x^7 \) and \( x^8 \) are equal:
\[ C_7 = C_8 \]
This gives us the equation:
\[ \binom{n}{7} \cdot 2^{n-7} \cdot \frac{1}{3^7} = \binom{n}{8} \cdot 2^{n-8} \cdot \frac{1}{3^8} \]

5. **Simplify the Equation**:
We can cancel \( \frac{1}{3^7} \) from both sides:
\[ \binom{n}{7} \cdot 2^{n-7} = \binom{n}{8} \cdot 2^{n-8} \cdot \frac{1}{3} \]
Rearranging gives:
\[ \binom{n}{7} \cdot 2^{n-7} = \frac{1}{3} \cdot \binom{n}{8} \cdot 2^{n-8} \]

6. **Use the Identity for Binomial Coefficients**:
We know that:
\[ \binom{n}{8} = \frac{n-7}{8} \cdot \binom{n}{7} \]
Substituting this into our equation:
\[ \binom{n}{7} \cdot 2^{n-7} = \frac{1}{3} \cdot \frac{n-7}{8} \cdot \binom{n}{7} \cdot 2^{n-8} \]

7. **Cancel \( \binom{n}{7} \) and Rearrange**:
Assuming \( \binom{n}{7} \neq 0 \):
\[ 2^{n-7} = \frac{1}{3} \cdot \frac{n-7}{8} \cdot 2^{n-8} \]
This simplifies to:
\[ 2 = \frac{n-7}{24} \]
Multiplying both sides by 24:
\[ 48 = n - 7 \]
Thus:
\[ n = 55 \]

### Conclusion:
The value of \( n \) is \( 55 \).