Question

ABCD is a convex quadrilateral with 3, 4, 5 and 6 points marked on its sides AB, BC, CD and DA respectively. Triangles are formed using these 18 points and original vertices of the quadrilateral. Number of such triangles that do not have any side or part of a side common with the quadrilateral are

• A. 342
• B. 461
• C. 422
• D. 384

Answer 1

The correct option is C

422

Case I: Only intermediate points are used
Using {AB, BC, CD} $\to 3\times 4\times 5=60$
Using {BC, CD, DA} $\to 4\times 5\times 6=120$
Using {CD, DA, AB} $\to 5\times 6\times 3=90$
Using {DA, AB, BC} $\to 6\times 3\times 4=72$
Total ways = 342

Case II: One vertex of quadrilateral is used
Using vertex A, other two points must be taken from intermediate points on BC and CD. No. of ways $=4\times 5=20.$
Similarly using vertex B, No. of ways $=5\times 6=30$
Using vertex C, No. of ways $=6\times 3=18$
Using vertex D, No. of ways $=3\times 4=12$
Total ways = 80
Combining case I and II, there are 422 possible triangles that have no side common with quadrilateral.