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:
The equation $a + b + 2c = 22$ is a linear Diophantine equation. Since $a, b, c$ are non-negative integers, we can fix the value of $c$ and find the number of possible non-negative integer solutions for $a$ and $b$ for each valid $c$.
Step 2: Key Formula or Approach:
For a fixed non-negative integer $k$, the number of non-negative integer solutions to the equation $a + b = k$ is given by $k + 1$.
Sum of an arithmetic progression: $S = \frac{n}{2}(first\_term + last\_term)$.
Step 3: Detailed Explanation:
Given $a + b + 2c = 22$.
Rewrite as: $a + b = 22 - 2c$.
Since $a \ge 0$ and $b \ge 0$, their sum must be non-negative:
$22 - 2c \ge 0 \implies 2c \le 22 \implies c \le 11$.
Since $c$ is a non-negative integer, the possible values for $c$ are $0, 1, 2, \dots, 11$.
For any chosen value of $c$, the equation becomes $a + b = k$, where $k = 22 - 2c$.
The number of solutions for $(a, b)$ is $k + 1 = (22 - 2c) + 1 = 23 - 2c$.
Total number of solutions $n(A)$ is the sum of solutions for all possible values of $c$:
$n(A) = \sum_{c=0}^{11} (23 - 2c)$
$n(A) = 23(12) - 2 \sum_{c=0}^{11} c$
$n(A) = 276 - 2 \left( \frac{11 \times 12}{2} \right)$
$n(A) = 276 - 132 = 144$.
Step 4: Final Answer:
The number of elements in set A is 144.