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:
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 \( 1 \).
Let a random variable \( X \) follow Poisson distribution such that \( P(X = 0) = 2P(X = 1) \). Then, P(X = 3) = ______
The probability distribution of a random variable \( X \) is given as follows. Then, \( P(X = 50) - \frac{P(X \leq 30)}{P(X \geq 20)} \) =