Q17
The number of distinct pairs of integers \((x, y)\) satisfying the inequalities \(x>y \geq 3\) and \(x+y<14\) is
Answer: 16
Key idea: Fix \(y\) and count the integers \(x\) allowed between \(y+1\) and the upper limit set by \(x+y<14\), i.e. \(x\le13-y\).
| \(y\) | range of \(x\) | count |
|---|---|---|
| 3 | 4 to 10 | 7 |
| 4 | 5 to 9 | 5 |
| 5 | 6 to 8 | 3 |
| 6 | 7 only | 1 |
| 7 | none (\(x\ge8\) but \(x\le6\)) | 0 |
Total \(=7+5+3+1=16\).
Tip: The counts fall by 2 each time \(y\) rises by 1, so the sum is a quick AP of odd numbers.
Hence, the answer is 16.