Question

What is the generating function for $ a_r $, where $ a_r $ is the number of integral solutions for the equation $$ a + b + c = r \quad \text{with} \quad 0 \leq a, b, c \leq 3? $$

• A. \( (1 + x + x^2 + x^3)^3 \)
• B. \( (1 + x + x^2 + x^3 + x^r)^r \)
• C. \( (1 + x + x^2 + x^3)^r \)
• D. \( (1 + x + x^2 + x^3 + x^r)^3 \)

Hint:

When dealing with generating functions for counting solutions to an equation, remember to represent the possible values for each variable using a sum of powers of \( x \). The total generating function is the product of these sums.

Answer 1

The Correct Option is A

We are tasked with finding the generating function \( a_r \), where \( a_r \) represents the number of integral solutions to the equation:

\( a + b + c = r \quad \text{with} \quad 0 \leq a, b, c \leq 3. \)

Step 1: Understanding the Problem

We are given that the variables \( a \), \( b \), and \( c \) can each take integer values from 0 to 3.
Therefore, the number of possible values for each variable is 4.

Step 2: Constructing the Generating Function for Each Variable

Each variable \( a \), \( b \), and \( c \) can take the following values: \( 0, 1, 2, 3 \).
Thus, the generating function for each variable is:

\( 1 + x + x^2 + x^3 \)

This represents the possible values of each variable, where:

• \( 1 \) represents \( a = 0 \),
• \( x \) represents \( a = 1 \),
• \( x^2 \) represents \( a = 2 \),
• \( x^3 \) represents \( a = 3 \).

Step 3: Combining the Generating Functions

Since the equation involves three variables \( a \), \( b \), and \( c \), the total generating function is the product of the individual generating functions:

\( (1 + x + x^2 + x^3)^3 \)

This is the generating function that encodes the number of integral solutions to the equation \( a + b + c = r \).

Step 4: Conclusion

The correct generating function is given by option 1:

\( (1 + x + x^2 + x^3)^3 \)

Thus, the correct answer is \( 1 \).