Find the number of terms in the expansion of
`((3a)/(4)+(b)/(2))^(9)`

Answer

To find the number of terms in the expansion of \(\left(\frac{3a}{4} + \frac{b}{2}\right)^{9}\), we can use the Binomial Theorem. According to the Binomial Theorem, the expansion of \((x + y)^n\) contains \(n + 1\) terms, where \(n\) is a non-negative integer.

### Step-by-Step Solution:

1. **Identify the expression**: We have the expression \(\left(\frac{3a}{4} + \frac{b}{2}\right)^{9}\).

2. **Recognize the form**: This expression is in the form of \((x + y)^n\) where:
- \(x = \frac{3a}{4}\)
- \(y = \frac{b}{2}\)
- \(n = 9\)

3. **Apply the Binomial Theorem**: The number of terms in the expansion of \((x + y)^n\) is given by \(n + 1\).

4. **Calculate the number of terms**: Substitute \(n = 9\) into the formula:
\[ \text{Number of terms} = n + 1 = 9 + 1 = 10 \]

5. **Conclusion**: Therefore, the number of terms in the expansion of \(\left(\frac{3a}{4} + \frac{b}{2}\right)^{9}\) is \(10\).

### Final Answer:
The number of terms in the expansion is **10**.