Find the number of terms in the expansion of `(2a + 3b + c)^5`

Answer

To find the number of terms in the expansion of \((2a + 3b + c)^5\), we can use the formula for the number of distinct terms in the expansion of a multinomial expression. The formula is given by:

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

where:
- \(n\) is the power to which the expression is raised,
- \(r\) is the number of different terms in the expression.

### Step-by-step Solution:

1. **Identify \(n\) and \(r\)**:
- In our expression \((2a + 3b + c)^5\), the power \(n = 5\).
- The terms inside the parentheses are \(2a\), \(3b\), and \(c\), which gives us \(r = 3\) (since there are three different terms).

2. **Apply the formula**:
- Substitute \(n\) and \(r\) into the formula:
\[ \text{Number of terms} = \binom{5 + 3 - 1}{3 - 1} \]
- Simplifying this gives:
\[ \text{Number of terms} = \binom{7}{2} \]

3. **Calculate \(\binom{7}{2}\)**:
- The binomial coefficient \(\binom{7}{2}\) is calculated as follows:
\[ \binom{7}{2} = \frac{7!}{2!(7-2)!} = \frac{7 \times 6}{2 \times 1} = \frac{42}{2} = 21 \]

4. **Conclusion**:
- Therefore, the number of distinct terms in the expansion of \((2a + 3b + c)^5\) is \(21\).

### Final Answer:
The number of terms in the expansion of \((2a + 3b + c)^5\) is **21**.