The number of feasible solutions \((x, y)\), with both x and y integers, of the constraints \(x + y \le 3\), \(x \le 2\), \(x \ge 0\), \(y \ge 0\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 9
A feasible solution is a point satisfying every constraint. Since \(x \le 2\) and \(x \ge 0\), the integer x is 0, 1 or 2, and for each x the integer y runs from 0 up to \(3 - x\).
\(x = 0\): \(y = 0, 1, 2, 3\) (4 points)
\(x = 1\): \(y = 0, 1, 2\) (3 points)
\(x = 2\): \(y = 0, 1\) (2 points)
Total \(= 4 + 3 + 2 = 9\)
Why the other options are incorrect
Option (1) 6 — Incorrect. This counts only the points with \(x + y \le 2\) and leaves out those on the line \(x + y = 3\).
Option (3) 10 — Incorrect. This also counts (3, 0), but \(x = 3 \gt 2\), so (3, 0) is not feasible.
Option (4) 7 — Incorrect. Counting column by column gives 4 + 3 + 2 = 9 points, not 7.