If n is an even integer and a, b, c are distinct
number, then the number of distinct terms in the expansion of
` ( a + b + c )^(n) + (a + b - c)^(n) `, is S

Options

A. `((n+2)/(2))^(2)`
B. `n + 2`
C. `(n+4)/(2)`
D. none of these

Explanation

To find the number of distinct terms in the expansion of \( (a + b + c)^n + (a + b - c)^n \), where \( n \) is an even integer and \( a, b, c \) are distinct numbers, we can follow these steps:

### Step 1: Understand the General Term
The general term in the expansion of \( (a + b + c)^n \) can be expressed using the multinomial theorem. The general term is given by:
\[ T = \frac{n!}{p!q!r!} a^p b^q c^r \]
where \( p + q + r = n \) and \( p, q, r \) are non-negative integers.

### Step 2: Expand Both Expressions
We will expand both \( (a + b + c)^n \) and \( (a + b - c)^n \).

1. **Expansion of \( (a + b + c)^n \)**:
\[ (a + b + c)^n = \sum_{p + q + r = n} \frac{n!}{p!q!r!} a^p b^q c^r \]

2. **Expansion of \( (a + b - c)^n \)**:
\[ (a + b - c)^n = \sum_{p + q + r' = n} \frac{n!}{p!q!r'!} a^p b^q (-c)^{r'} \]
Here, \( r' \) is the exponent of \(-c\), which will affect the sign of the terms.

### Step 3: Combine the Two Expansions
Now, we need to consider the combined expression:
\[ (a + b + c)^n + (a + b - c)^n \]
When we add these two expansions, terms with odd powers of \( c \) will cancel out because they will have opposite signs, while terms with even powers of \( c \) will add up.

### Step 4: Determine the Distinct Terms
Since \( n \) is even, we can let \( r = 2k \) (where \( k \) is a non-negative integer). The distinct terms will be of the form:
\[ a^p b^q c^{2k} \]
where \( p + q + 2k = n \).

### Step 5: Count the Distinct Terms
To count the distinct terms, we need to find the number of non-negative integer solutions to the equation \( p + q + k = \frac{n}{2} \).

Using the stars and bars combinatorial method, the number of solutions is given by:
\[ \text{Number of solutions} = \binom{\frac{n}{2} + 2}{2} \]
This accounts for \( p, q, \) and \( k \).

### Step 6: Final Expression for Distinct Terms
Thus, the total number of distinct terms \( S \) in the expansion is:
\[ S = \frac{(n/2 + 2)(n/2 + 1)}{2} \]

### Conclusion
The final result for the number of distinct terms in the expansion of \( (a + b + c)^n + (a + b - c)^n \) is:
\[ S = \frac{n + 2}{2} \]