The number of distinct pairs of integers $(x, y)$ satisfying the inequalities $x > y \geq 3$ and $x + y < 14$ is

Entered answer:

✅ Correct Answer: 16

The three conditions are: $x > y$ , $y \geq 3$ , and $x + y < 14$ .

For any fixed $y$ , the valid range for $x$ is: $y < x < 14 - y$

When $y = 3$ : $x$ ranges from $4$ to $10$ , so $x \in \{4, 5, 6, 7, 8, 9, 10\}$ — 7 pairs

When $y = 4$ : $x$ ranges from $5$ to $9$ , so $x \in \{5, 6, 7, 8, 9\}$ — 5 pairs

When $y = 5$ : $x$ ranges from $6$ to $8$ , so $x \in \{6, 7, 8\}$ — 3 pairs

When $y = 6$ : $x$ ranges from $7$ to $7$ , so $x \in \{7\}$ — 1 pair

When $y = 7$ : $x > 7$ and $x < 7$ is impossible — 0 pairs

For $y \geq 7$ , no valid $x$ exists, so we stop here.

Total pairs $= 7 + 5 + 3 + 1 = 16$

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CAT Past Year Questions (Topic-Wise)

The number of distinct pairs of integers $(x, y)$ satisfying the inequalities $x > y \geq 3$ and $x + y < 14$ is