Find the number of distinct terms in the expansion
`(x + y -z)^16`

Answer

To find the number of distinct terms in the expansion of \((x + y - z)^{16}\), we can use the formula for the number of distinct terms in the expansion of \((x_1 + x_2 + x_3 + \ldots + x_r)^n\), which is given by:

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

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

### Step 1: Identify the values of \(n\) and \(r\)
In our case:
- The expression is \((x + y - z)^{16}\).
- The power \(n = 16\).
- The number of distinct variables \(r = 3\) (which are \(x\), \(y\), and \(-z\)).

### Step 2: Apply the formula
Now we can substitute the values of \(n\) and \(r\) into the formula:

\[ \text{Number of distinct terms} = \binom{16 + 3 - 1}{3 - 1} \]

### Step 3: Simplify the expression
This simplifies to:

\[ \text{Number of distinct terms} = \binom{18}{2} \]

### Step 4: Calculate the binomial coefficient
Now we calculate \(\binom{18}{2}\):

\[ \binom{18}{2} = \frac{18 \times 17}{2 \times 1} = \frac{306}{2} = 153 \]

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