In the expansion of `(1 + x)^(n) (1 + y)^(n) (1 + z)^(n)` , the sum of the co-efficients of the terms of degree `r` is

Options

A. `.^(n^(3))C_(r)`
B. `.^(n)C_(r^(3))`
C. `.^(3n)C_(r)`
D. `3.^(2n)C_(r)`

Answer

To find the sum of the coefficients of the terms of degree \( r \) in the expansion of \( (1 + x)^{n} (1 + y)^{n} (1 + z)^{n} \), we can follow these steps:

### Step 1: Understand the Expansion
The expression \( (1 + x)^{n} (1 + y)^{n} (1 + z)^{n} \) can be expanded using the binomial theorem. Each binomial expansion contributes terms of varying degrees.

### Step 2: Identify the Degree of Terms
The degree of a term in the expansion is the sum of the powers of \( x \), \( y \), and \( z \). For example, a term like \( x^a y^b z^c \) has a degree \( a + b + c \).

### Step 3: Count the Coefficients
To find the sum of the coefficients of the terms of degree \( r \), we need to consider all combinations of \( x \), \( y \), and \( z \) that add up to \( r \). This can be represented as selecting \( r \) items from three groups (corresponding to \( x \), \( y \), and \( z \)).

### Step 4: Use the Multinomial Theorem
The total number of ways to choose \( r \) items from three groups (where each group can contribute from 0 to \( r \)) can be calculated using the multinomial coefficient. Specifically, we can express this as:

\[ \sum_{a + b + c = r} \frac{r!}{a! b! c!} \]

### Step 5: Substitute into the Binomial Expansion
Since we are dealing with three identical expansions of \( (1 + x)^{n} \), we can rewrite the expression as:

\[ (1 + x)^{3n} \]

### Step 6: Find the Coefficient of \( x^r \)
The coefficient of \( x^r \) in the expansion of \( (1 + x)^{3n} \) is given by:

\[ \binom{3n}{r} \]

### Final Answer
Thus, the sum of the coefficients of the terms of degree \( r \) in the expansion of \( (1 + x)^{n} (1 + y)^{n} (1 + z)^{n} \) is:

\[ \binom{3n}{r} \]