Question
$$ABCD$$ is a convex quadrilateral. 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.
None of these
Answer :
None of these
Solution :
The number of triangles with vertices on sides $$AB,BC,CD = {\,^3}{C_1} \times {\,^4}{C_1} \times {\,^5}{C_1}.$$
Similarly for other cases.
∴ the total number of triangles
$$ = {\,^3}{C_1} \times {\,^4}{C_1} \times {\,^5}{C_1} + {\,^3}{C_1} \times {\,^4}{C_1} \times {\,^6}{C_1} + {\,^3}{C_1} \times {\,^5}{C_1} \times {\,^6}{C_1} + {\,^4}{C_1} \times {\,^5}{C_1} \times {\,^6}{C_1} = 342.$$
The number of triangles with vertices on sides $$AB,BC,CD = {\,^3}{C_1} \times {\,^4}{C_1} \times {\,^5}{C_1}.$$
Similarly for other cases.
∴ the total number of triangles
$$ = {\,^3}{C_1} \times {\,^4}{C_1} \times {\,^5}{C_1} + {\,^3}{C_1} \times {\,^4}{C_1} \times {\,^6}{C_1} + {\,^3}{C_1} \times {\,^5}{C_1} \times {\,^6}{C_1} + {\,^4}{C_1} \times {\,^5}{C_1} \times {\,^6}{C_1} = 342.$$