If the coefficients of `x^2a n d\ x^3` in the expansion o `(3+a x)^9` are the same, then the value of `a` is `-7/9` b. `-9/7` c. `7/9` d. `9/7`

Explanation

To solve the problem, we need to find the value of \( a \) such that the coefficients of \( x^2 \) and \( x^3 \) in the expansion of \( (3 + ax)^9 \) are equal.

### Step-by-Step Solution:

1. **Identify the General Term**:
The general term in the binomial expansion of \( (3 + ax)^n \) is given by:
\[ T_k = \binom{n}{k} (3)^{n-k} (ax)^k \]
For our case, \( n = 9 \).

2. **Find the Coefficient of \( x^2 \)**:
The coefficient of \( x^2 \) corresponds to \( k = 2 \):
\[ T_2 = \binom{9}{2} (3)^{9-2} (ax)^2 = \binom{9}{2} (3)^7 a^2 \]
The coefficient of \( x^2 \) is:
\[ C_2 = \binom{9}{2} \cdot 3^7 \cdot a^2 \]

3. **Find the Coefficient of \( x^3 \)**:
The coefficient of \( x^3 \) corresponds to \( k = 3 \):
\[ T_3 = \binom{9}{3} (3)^{9-3} (ax)^3 = \binom{9}{3} (3)^6 (ax)^3 \]
The coefficient of \( x^3 \) is:
\[ C_3 = \binom{9}{3} \cdot 3^6 \cdot a^3 \]

4. **Set the Coefficients Equal**:
Since the coefficients of \( x^2 \) and \( x^3 \) are the same, we have:
\[ \binom{9}{2} \cdot 3^7 \cdot a^2 = \binom{9}{3} \cdot 3^6 \cdot a^3 \]

5. **Simplify the Equation**:
We can simplify this equation:
\[ \binom{9}{2} \cdot 3^7 \cdot a^2 = \binom{9}{3} \cdot 3^6 \cdot a^3 \]
Dividing both sides by \( 3^6 \):
\[ \binom{9}{2} \cdot 3 \cdot a^2 = \binom{9}{3} \cdot a^3 \]

6. **Calculate the Binomial Coefficients**:
We know:
\[ \binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36 \]
\[ \binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 \]
Substituting these values into the equation gives:
\[ 36 \cdot 3 \cdot a^2 = 84 \cdot a^3 \]
Simplifying further:
\[ 108 a^2 = 84 a^3 \]

7. **Rearranging the Equation**:
Rearranging gives:
\[ 84 a^3 - 108 a^2 = 0 \]
Factoring out \( a^2 \):
\[ a^2 (84 a - 108) = 0 \]

8. **Solving for \( a \)**:
This gives us two solutions:
\[ a^2 = 0 \quad \text{or} \quad 84 a - 108 = 0 \]
The first gives \( a = 0 \) (not relevant here), and the second gives:
\[ 84 a = 108 \implies a = \frac{108}{84} = \frac{9}{7} \]

### Final Answer:
Thus, the value of \( a \) is \( \frac{9}{7} \).