QJEEM21Ma4e74: JEE Main 2021 Mathematics - Class 11 JEEM21Ma4e74 Solution: Consider a rectangle ABCD having 5, 7, 6, 9 points in the…

Question

Verified Solution • 2025–26 Session moderate

Consider a rectangle ABCD having 5, 7, 6, 9 points in the interior of the line segments AB, CD, BC, DA respectively. Let

α

be the number of triangles having these points from different sides as vertices and

β

be the number of quadrilaterals having these points from different sides as vertices. Then (

β

-

α

) is equal to :

Accepted Answer

α =^{6} C_{1}^{7} C_{1}^{9} C_{1} +^{5} C_{1}^{7} C_{1}^{9} C_{1} +^{5} C_{1}^{6} C_{1}^{9} C_{1} +^{5} C_{1}^{6} C_{1}^{7} C_{1}

= 378 + 315 + 270 + 210 = 1173

β =^{5} C_{1}^{6} C_{1}^{7} C_{1}^{9} C_{1} = 1890

∴

β - α = 1890 - 1173 = 717

Step-by-Step Verified Solution