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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Permutation and Combination

NDA 2024 · Mathematics · Permutation and Combination

A triangle PQR is such that 3 points lie on the side PQ, 4 points...

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

Options

  1. A205
  2. B206
  3. C215
  4. D220

Correct answer

A. 205

Step-by-step solution

Total no. of trianglesformedtriangles formed=12C3−(3C3+4C3+5C3)=12×11×10×9!9!×3!−(1+4+10)=12×11×103×2−15=2×110−15=220−15=205\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}$$
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