Question

If PQRS is a convex quadrilateral with 3, 4, 5 and 6 points marked on sides PQ, QR, RS and PS respectively, then the number of triangles with vertices on different sides is

• A. 220
• B. 270
• C. 342
• D. 282

Answer 1

Answer: (C)

Total number of triangles = Number of triangles with vertices on sides PQ, OR and RS+ Number of triangles with vertices on sides OR, 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}}{{\times }^{4}}{{C}_{1}}{{\times }^{5}}{{C}_{1}}{{\times }^{4}}{{C}_{1}}{{\times }^{5}}{{C}_{1}}{{\times }^{6}}{{C}_{1}} $ $ +{{\,}^{5}}{{C}_{1}}{{\times }^{6}}\,{{C}_{1}}{{\times }^{3}}\,{{C}_{1}}{{+}^{6}}{{C}_{1}}{{\times }^{3}}\,{{C}_{1}}{{\times }^{4}}{{C}_{1}} $ $ =60+120+90+72=342 $