The coefficient of `x^(n) y^(n)` in the expansion of
`[(1 + x)(1+y) (x +y)]^(n)` , is

Options

A. `sum_(r=0)^(n) C_(r)""^(2)`
B. `sum_(r=0)^(n) C_(r+2)^(2)`
C. `sum_(r=0)^(n) C_(r+3)^(2)`
D. `sum_(r=0)^(n) C_(r)^(3)`

Answer

To find the coefficient of \( x^n y^n \) in the expansion of \( [(1 + x)(1 + y)(x + y)]^n \), we will follow these steps:

### Step 1: Expand Each Factor
We start by expanding each factor in the expression \( (1 + x)(1 + y)(x + y) \).

1. **Expand \( (1 + x) \)**:
\[ (1 + x) = 1 + x \]

2. **Expand \( (1 + y) \)**:
\[ (1 + y) = 1 + y \]

3. **Expand \( (x + y) \)**:
\[ (x + y) = x + y \]

### Step 2: Combine the Expansions
Now, we can combine these expansions:
\[ (1 + x)(1 + y)(x + y) = (1 + x)(1 + y)(x + y) \]
This will give us a polynomial in \( x \) and \( y \).

### Step 3: Use the Multinomial Expansion
We need to raise the combined expression to the power of \( n \):
\[ [(1 + x)(1 + y)(x + y)]^n \]
Using the multinomial theorem, we can express this as:
\[ \sum_{a+b+c=n} \frac{n!}{a!b!c!} (1^a)(x^b)(y^c) \]
where \( a, b, c \) are the number of times each term is chosen from the expansion.

### Step 4: Identify Terms for \( x^n y^n \)
We are interested in the coefficient of \( x^n y^n \). This occurs when:
- The total degree of \( x \) is \( n \)
- The total degree of \( y \) is \( n \)

### Step 5: Find Coefficients
To find the coefficient of \( x^n y^n \), we need to consider the contributions from each part of the expansion:
- From \( (1 + x) \), we can choose \( x \) a total of \( b \) times.
- From \( (1 + y) \), we can choose \( y \) a total of \( c \) times.
- From \( (x + y) \), we can choose \( x \) and \( y \) in such a way that their total degree sums to \( n \).

### Step 6: Summation of Coefficients
The coefficient of \( x^n y^n \) in the expansion can be expressed as:
\[ \sum_{r=0}^{n} \binom{n}{r}^3 \]
This represents the sum of the cubes of the binomial coefficients.

### Final Answer
Thus, the coefficient of \( x^n y^n \) in the expansion of \( [(1 + x)(1 + y)(x + y)]^n \) is:
\[ \sum_{r=0}^{n} \binom{n}{r}^3 \]