Question:
Consider a rectangle ABCD having 5, 7, 6, 9 points in the interior of the line segments AB, CD, BC, DA respectively. Let \(\alpha\) be the number of triangles having these points from different sides as vertices and \(\beta\) be the number of quadrilaterals having these points from different sides as vertices. Then \((\beta - \alpha)\) is equal to :
Consider a rectangle ABCD having 5, 7, 6, 9 points in the interior of the line segments AB, CD, BC, DA respectively. Let \(\alpha\) be the number of triangles having these points from different sides as vertices and \(\beta\) be the number of quadrilaterals having these points from different sides as vertices. Then \((\beta - \alpha)\) is equal to :
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A triangle needs points from three different sides; a quadrilateral needs one point from each of the four sides.
Updated On: Nov 4, 2026
- 717
- 795
- 1890
- 1173
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The Correct Option is A
Solution and Explanation
Points on the four sides: 5, 7, 6, 9.
Quadrilaterals (one point from each side): \(\beta = 5\times7\times6\times9 = 1890\).
Triangles (points from three different sides): \(\alpha = 5\cdot7\cdot6 + 5\cdot7\cdot9 + 5\cdot6\cdot9 + 7\cdot6\cdot9 = 210 + 315 + 270 + 378 = 1173\).
\(\beta - \alpha = 1890 - 1173 = 717\).
Answer: (1) 717
Quadrilaterals (one point from each side): \(\beta = 5\times7\times6\times9 = 1890\).
Triangles (points from three different sides): \(\alpha = 5\cdot7\cdot6 + 5\cdot7\cdot9 + 5\cdot6\cdot9 + 7\cdot6\cdot9 = 210 + 315 + 270 + 378 = 1173\).
\(\beta - \alpha = 1890 - 1173 = 717\).
Answer: (1) 717
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