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

Options

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

Answer

To find the number of terms in the expansion of \((x + y + z)^n\), we can use the formula for the number of distinct terms in the expansion of a multinomial expression. Here’s how we can derive the solution step by step:

### Step 1: Understand the Multinomial Expansion
The expansion of \((x + y + z)^n\) involves terms of the form \(x^a y^b z^c\) where \(a + b + c = n\). Each term corresponds to a unique combination of the powers of \(x\), \(y\), and \(z\).

### Step 2: Use the Stars and Bars Theorem
To find the number of non-negative integer solutions to the equation \(a + b + c = n\), we can use the "stars and bars" theorem. This theorem states that the number of ways to distribute \(n\) identical objects (stars) into \(r\) distinct groups (variables) is given by the formula:

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

### Step 3: Identify \(r\) in Our Case
In our case, we have three variables \(x\), \(y\), and \(z\), so \(r = 3\).

### Step 4: Substitute into the Formula
Now, substituting \(n\) for the total number of stars and \(r = 3\) into the formula gives:

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

### Step 5: Calculate the Binomial Coefficient
The binomial coefficient \(\binom{n + 2}{2}\) can be calculated using the formula:

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

### Final Answer
Thus, the number of terms in the expansion of \((x + y + z)^n\) is:

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