The number of terms in the expansion of `(2x+3y-4z)^n` is

Options

A. n+1
B. n+3
C. `((n+1)(n+2))/(2) `
D. none of these

Answer

To find the number of terms in the expansion of \((2x + 3y - 4z)^n\), we can use the formula for the number of terms in the expansion of a multinomial expression. The formula states that the number of terms in the expansion of \((x_1 + x_2 + \ldots + x_p)^n\) is given by:

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

where \(p\) is the number of different variables in the expression.

### Step-by-step Solution:

1. **Identify the variables**: In the expression \((2x + 3y - 4z)^n\), we have three different variables: \(2x\), \(3y\), and \(-4z\). Thus, \(p = 3\).

2. **Apply the formula**: We substitute \(n\) (the exponent) and \(p\) (the number of variables) into the formula:

\[ \text{Number of terms} = \binom{n + p - 1}{p - 1} = \binom{n + 3 - 1}{3 - 1} = \binom{n + 2}{2} \]

3. **Simplify the binomial coefficient**: The binomial coefficient \(\binom{n + 2}{2}\) can be expressed as:

\[ \binom{n + 2}{2} = \frac{(n + 2)(n + 1)}{2} \]

4. **Conclusion**: Therefore, the number of terms in the expansion of \((2x + 3y - 4z)^n\) is:

\[ \frac{(n + 2)(n + 1)}{2} \]

### Final Answer:
The number of terms in the expansion of \((2x + 3y - 4z)^n\) is \(\frac{(n + 2)(n + 1)}{2}\).

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