Find the number of terms in the expansion of
`(2x+3y+z)^(7)`

Answer

To find the number of terms in the expansion of \((2x + 3y + z)^7\), we can use the formula for the number of distinct terms in the expansion of \((x_1 + x_2 + \ldots + x_r)^n\), which is given by:

\[ \text{Number of terms} = \binom{n + r - 1}{r - 1} \]

where \(n\) is the exponent and \(r\) is the number of different variables.

### Step-by-step Solution:

1. **Identify the values of \(n\) and \(r\)**:
- In the expression \((2x + 3y + z)^7\):
- The exponent \(n = 7\).
- The number of different variables \(r\) (which are \(2x\), \(3y\), and \(z\)) is \(3\).

2. **Apply the formula**:
- Substitute \(n\) and \(r\) into the formula:
\[ \text{Number of terms} = \binom{n + r - 1}{r - 1} = \binom{7 + 3 - 1}{3 - 1} \]

3. **Simplify the expression**:
- Calculate \(n + r - 1\) and \(r - 1\):
\[ n + r - 1 = 7 + 3 - 1 = 9 \]
\[ r - 1 = 3 - 1 = 2 \]
- Thus, we have:
\[ \text{Number of terms} = \binom{9}{2} \]

4. **Calculate \(\binom{9}{2}\)**:
- The binomial coefficient \(\binom{9}{2}\) is calculated as follows:
\[ \binom{9}{2} = \frac{9!}{2!(9-2)!} = \frac{9 \times 8}{2 \times 1} = \frac{72}{2} = 36 \]

5. **Conclusion**:
- Therefore, the number of terms in the expansion of \((2x + 3y + z)^7\) is \(36\).

### Final Answer:
The number of terms in the expansion of \((2x + 3y + z)^7\) is **36**.