Question

ABCD is a convex quadrilateral and 3, 4, 5, and 6 points are marked on the sides AB, BC, CD, and DA, respectively. The number of triangles with vertices on different sides is a. `270` b. `220` c. `282` d. `342`
A. 220
B. 270
C. 282
D. 342

Answer 1

`=` Number of triangales with vertices on sides PQ, QR, and RS
+ Number of triangles with vertices on sides QR, RS and PS
+ Number of triangles with vertices on sides RS, PS and PQ
+ Number of triangles with vertices on sides PS, PQ and QR
`=""^(3)C_(1)xx""^(4)C_(1)xx""^(5)C_(1)xx""^(4)C_(1)xx""^(5)C_(1)xx""^(6)C_(1)+""^(5)C_(1)xx""^(6)C_(1)xx""^(3)C_(1)+""^(6)C_(1)xx""^(3)C_(1)xx""^(4)C_(1)`
`=60+120+90+72+=342`.