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. $342$

Answer 1

The correct option is D $342$

The number 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 number of triangles is
${}^3{C}_1{\times }^4{C}_1{\times }^5{C}_1{+}^3{C}_1{\times }^4{C}_1{\times }^6{C}_1{+}^3{C}_1{\times }^5{C}_1{\times }^6{C}_1{+}^4{C}_1{\times }^5{C}_1{\times }^6{C}_1$
$=342$.