The number of terms in `(2x + 3y + z - w)^20` is

Answer

To find the number of terms in the expansion of \((2x + 3y + z - w)^{20}\), we can use the formula for the number of terms in the expansion of a multinomial expression.

### Step-by-Step Solution:

1. **Identify the number of variables**:
The expression \(2x + 3y + z - w\) contains 4 variables: \(x\), \(y\), \(z\), and \(w\).

2. **Determine the exponent**:
The exponent of the expression is 20.

3. **Use the formula for the number of terms**:
The number of terms in the expansion of \((x_1 + x_2 + x_3 + \ldots + x_r)^n\) is given by the formula:
\[ \text{Number of terms} = \binom{n + r - 1}{r - 1} \]
where \(n\) is the exponent and \(r\) is the number of variables.

4. **Substitute the values into the formula**:
Here, \(n = 20\) and \(r = 4\) (since we have 4 variables). Thus, we substitute these values into the formula:
\[ \text{Number of terms} = \binom{20 + 4 - 1}{4 - 1} = \binom{23}{3} \]

5. **Calculate \(\binom{23}{3}\)**:
Using the formula for combinations:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
we have:
\[ \binom{23}{3} = \frac{23!}{3!(23-3)!} = \frac{23!}{3! \cdot 20!} \]

6. **Simplify the factorial expression**:
We can simplify this as follows:
\[ \binom{23}{3} = \frac{23 \times 22 \times 21 \times 20!}{3! \times 20!} \]
The \(20!\) cancels out:
\[ = \frac{23 \times 22 \times 21}{3!} \]

7. **Calculate \(3!\)**:
\(3! = 3 \times 2 \times 1 = 6\).

8. **Final calculation**:
Now we compute:
\[ = \frac{23 \times 22 \times 21}{6} \]
First, calculate \(23 \times 22 = 506\).
Then, \(506 \times 21 = 10626\).
Finally, divide by 6:
\[ = \frac{10626}{6} = 1771 \]

Thus, the number of terms in the expansion of \((2x + 3y + z - w)^{20}\) is **1771**.