Q66 / 170 · Manipal (MET) 2015 · Mathematics

If $P Q R S$ is a convex quadrilateral with $3,4,5$ and 6 points marked on sides $P Q, Q R, R S$ and PS respectively. Then, the number of triangles with vertices on different sides is

  • A
    $220$
  • B
    $270$
  • C
    $282$
  • Correct answer
    D
    $342$

Solution

Total number of triangles $=$ Number of triangles with vertices on sides ( $P Q, Q R, R S+O R, R S$, $P S+R S, P S, P Q+P S, P Q, Q R)$ $\begin{array}{r}={ }^3 C_1 \times{ }^4 C_1 \times{ }^5 C_1+{ }^4 C_1 \times{ }^5 C_1 \times{ }^6 C_1+{ }^5 C_1 \times{ }^6 C_1 \\ \times{ }^3 C_1+{ }^6 C_1 \times{ }^3 C_1 \times{ }^4 C_1\end{array}$ $=60+120+90+72=342$

Hence, the correct answer is option: 'D'.

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