Question

The number of non-negative integer solutions of the equation \[ a + b + 2c = 22 \] is:

• A. \(124\)
• B. \(144\)
• C. \(135\)
• D. \(136\)

Answer 1

The Correct Option is A

Concept: Use the stars and bars method for counting non-negative integer solutions. For an equation \[ x_1 + x_2 + \dots + x_n = k \] the number of non-negative solutions is \[ \binom{k+n-1}{n-1} \] Step 1: Fix the value of \(c\). From the equation \[ a + b + 2c = 22 \] \[ a + b = 22 - 2c \] Since \(a,b,c \ge 0\), \[ 22 - 2c \ge 0 \] \[ c = 0,1,2,\dots,11 \] Step 2: Count solutions for each \(c\). For a fixed \(c\), \[ a + b = 22 - 2c \] Number of solutions: \[ = (22 - 2c) + 1 \] Step 3: Sum for all values of \(c\). \[ \sum_{c=0}^{11} (23 - 2c) \] \[ = 23 + 21 + 19 + \dots + 1 \] This is an arithmetic progression with \(12\) terms. \[ S = \frac{n}{2}(a_1 + a_n) \] \[ S = \frac{12}{2}(23 + 1) \] \[ S = 6 \times 24 \] \[ S = 144 \]