Question

Let \(A = \{ (a, b, c) : a, b, c \text{ are non-negative integers and } a + b + 2c = 22 \}\). Then \(n(A)\) is equal to:

• A. 121
• B. 124
• C. 144
• D. 169

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We need to find the number of non-negative integer solutions to a linear Diophantine equation. We can iterate through possible values of \(c\) and use the stars and bars method for the remaining variables.

Step 2: Key Formula or Approach:
For \(a + b = N\), the number of non-negative integer solutions is \(N + 1\).

Step 3: Detailed Explanation:
\(a + b = 22 - 2c\). Since \(a, b \geq 0\), we must have \(22 - 2c \geq 0 \implies c \in \{0, 1, 2, \dots, 11\}\).
If \(c = 0, a + b = 22 \implies 23\) solutions.
If \(c = 1, a + b = 20 \implies 21\) solutions.
If \(c = 2, a + b = 18 \implies 19\) solutions.
...
If \(c = 11, a + b = 0 \implies 1\) solution.
Total solutions \(= 1 + 3 + 5 + \dots + 23\).
This is a sum of the first 12 odd numbers.
Sum \(= 12^2 = 144\).

Step 4: Final Answer:
\(n(A) = 144\).