question_answer

If ABCD is a convex quadrilateral and 3, 4, 5 and 6 points are marked on the sides AB, BC, CD and DA respectively. The no. of triangles with vertices on different sides is -

A) 270          

B)                    220

C) 282                  

D)        342

Correct Answer: D

Solution :

[d] The no. of triangles with vertices on sides AB, BC, CD is \[^{3}{{C}_{1}}{{\times }^{4}}{{C}_{1}}{{\times }^{5}}{{C}_{1}}\] Similarly, for other cases, the total no. of triangles is \[^{3}{{C}_{1}}{{\times }^{4}}{{C}_{1}}{{\times }^{5}}{{C}_{1}}{{+}^{3}}{{C}_{1}}{{\times }^{4}}{{C}_{1}}{{\times }^{6}}{{C}_{1}}{{+}^{3}}{{C}_{1}}{{\times }^{5}}{{C}_{1}}{{\times }^{6}}{{C}_{1}}\]\[{{+}^{4}}{{C}_{1}}{{\times }^{5}}{{C}_{1}}{{\times }^{6}}{{C}_{1}}=342\]