JEE MAIN - Mathematics (2021 - 16th March Evening Shift - No. 5)
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 :
717
795
1890
1173
Explanation
$$\alpha = {}^6{C_1}{}^7{C_1}{}^9{C_1} + {}^5{C_1}{}^7{C_1}{}^9{C_1} + {}^5{C_1}{}^6{C_1}{}^9{C_1} + {}^5{C_1}{}^6{C_1}{}^7{C_1} $$
$$= 378 + 315 + 270 + 210 = 1173$$
$$\beta = {}^5{C_1}{}^6{C_1}{}^7{C_1}{}^9{C_1} = 1890$$
$$ \therefore $$ $$ \beta - \alpha = 1890 - 1173 = 717$$
$$= 378 + 315 + 270 + 210 = 1173$$
$$\beta = {}^5{C_1}{}^6{C_1}{}^7{C_1}{}^9{C_1} = 1890$$
$$ \therefore $$ $$ \beta - \alpha = 1890 - 1173 = 717$$
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