Q156 / 270 · NDA 2024 (Phase 1) · Mathematics

A triangle PQR is such that 3 points lie on the side $\mathrm{PQ}, 4$ points on QR and 5 points on $R P$ respectively. Triangles are constructed using these points as vertices. What is the number of triangles so formed?

  • Correct answer
    A
    205
  • B
    206
  • C
    215
  • D
    220

Solution

Total no. of triangles formed $\begin{aligned} & ={ }^{12} \mathrm{C}_3-\left({ }^3 \mathrm{C}_3+{ }^4 \mathrm{C}_3+{ }^5 \mathrm{C}_3\right) \\ & =\frac{12 \times 11 \times 10 \times 9!}{9!\times 3!}-(1+4+10) \\ & =\frac{12 \times 11 \times 10}{3 \times 2}-15 \\ & =2 \times 110-15 \\ & =220-15=205 \end{aligned}$

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

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