No. of distinct terms in `(x + y - z)^16` is

Options

A. 154
B. 126
C. 133
D. 153

Answer

To find the number of distinct terms in the expansion of \((x + y - z)^{16}\), we can use the multinomial theorem. The general formula for the number of distinct terms in the expansion of \((a_1 + a_2 + ... + a_r)^n\) is given by:

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

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

### Step-by-Step Solution:

1. **Identify \(n\) and \(r\)**:
- Here, \(n = 16\) (the exponent) and \(r = 3\) (the variables \(x\), \(y\), and \(-z\)).

2. **Apply the formula**:
- We need to calculate:
\[ \binom{n + r - 1}{r - 1} = \binom{16 + 3 - 1}{3 - 1} = \binom{18}{2} \]

3. **Calculate \(\binom{18}{2}\)**:
- The formula for combinations is:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
- Thus,
\[ \binom{18}{2} = \frac{18!}{2!(18-2)!} = \frac{18!}{2! \cdot 16!} \]
- This simplifies to:
\[ \binom{18}{2} = \frac{18 \times 17}{2 \times 1} = \frac{306}{2} = 153 \]

4. **Conclusion**:
- Therefore, the number of distinct terms in the expansion of \((x + y - z)^{16}\) is **153**.

### Final Answer:
The number of distinct terms in \((x + y - z)^{16}\) is **153**.