ABCD is a convex quadrilateral. 3,4,5 and 6 points are marked on the sides AB , BC , CD and DA respectively. Find the number of triangles with vertices on different sides.

Question

ABCD is a convex quadrilateral. 3,4,5 and 6 points are marked on the sides AB , BC , CD and DA respectively. Find the number of triangles with vertices on different sides. (A) 215 (B) 342 (C) 225 (D) 424

Tardigrade Admin · Accepted Answer

The number of triangles with vertices on sides

A B, BC, C D =^{3} C_{1} \times^{4} C_{1} \times^{5} C_{1}

Similarly for other cases

∴

The total number of triangles

=^{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

Q. ABCD is a convex quadrilateral. 3,4,5 and 6 points are marked on the sides AB,BC,CD and DA respectively. Find the number of triangles with vertices on different sides.

215

342

225

424

The number of triangles with vertices on sides AB,BC,CD=3C1​×4C1​×5C1​ Similarly for other cases ∴ The total number of triangles =3C1​×4C1​×5C1​+3C1​×4C1​×6C1​ +3C1​×5C1​×6C1​+4C1​×5C1​×6C1​=342

Q. $A BC D$ is a convex quadrilateral. $3, 4, 5$ and $6$ points are marked on the sides $A B, BC, C D$ and DA respectively. Find the number of triangles with vertices on different sides.