Question

Number of non-negative integral solutions of the equation \[ a+b+2c=22 \] is

• A. \(144\)
• B. \(121\)
• C. \(168\)
• D. \(99\)

Hint:

Number of non-negative solutions of \(a+b=n\) is \(n+1\).

Answer 1

The Correct Option is A

Concept: Fix \(c\) and count solutions for \(a+b\) using combinations.
Step 1: Rewrite equation \[ a+b=22-2c \] Since \(a,b,c\ge0\), \[ 22-2c\ge0 \] \[ c=0,1,2,\ldots,11 \]
Step 2: Count solutions for each \(c\) Number of solutions of \(a+b=k\): \[ k+1 \] Thus \[ (22-2c)+1=23-2c \]
Step 3: Total solutions \[ \sum_{c=0}^{11}(23-2c) \] \[ =23+21+19+\cdots+1 \] This is sum of first 12 odd numbers. \[ =12^2 \] \[ =144 \]