Find the number of terms in the expansion of `(x - 2y + 3z)^10`

Answer

To find the number of terms in the expansion of \((x - 2y + 3z)^{10}\), we can use the formula for the number of terms in the expansion of \((a + b + c)^n\), which is given by:

\[ \text{Number of terms} = n + k - 1 \choose k - 1 \]

where \(n\) is the power to which the expression is raised, and \(k\) is the number of different variables in the expression.

### Step-by-step Solution:

1. **Identify the variables and the power**:
- In the expression \((x - 2y + 3z)^{10}\), we have three variables: \(x\), \(-2y\), and \(3z\).
- The power \(n\) is \(10\).

2. **Count the number of variables**:
- The number of distinct variables \(k\) is \(3\) (which are \(x\), \(y\), and \(z\)).

3. **Apply the formula**:
- Substitute \(n = 10\) and \(k = 3\) into the formula:
\[ \text{Number of terms} = 10 + 3 - 1 \choose 3 - 1 = 12 \choose 2 \]

4. **Calculate \(12 \choose 2\)**:
- The binomial coefficient \(12 \choose 2\) is calculated as follows:
\[ 12 \choose 2 = \frac{12!}{2!(12-2)!} = \frac{12 \times 11}{2 \times 1} = \frac{132}{2} = 66 \]

5. **Conclusion**:
- Therefore, the number of terms in the expansion of \((x - 2y + 3z)^{10}\) is \(66\).