Solve the Following Questions


Let $\mathrm{P}_{1}, \mathrm{P}_{2}, \ldots, \mathrm{P}_{15}$ be 15 points on a circle. The number of distinct triangles formed by points $P_{i}, P_{j}, P_{k}$ such that $i+j+k \neq 15$, is :

  1. 12

  2. 419

  3. 443

  4. 455

Correct Option: , 3


Total Number of Triangles $={ }^{15} \mathrm{C}_{3}$

$\mathrm{i}+\mathrm{j}+\mathrm{k}=15$ (Given)

Number of Possible triangles using the vertices $\mathrm{P}_{\mathrm{i}}, \mathrm{P}_{\mathrm{j}}$, $P_{k}$ such that $i+j+k \neq 15$ is equal to ${ }^{15} C_{3}-12=443$

Option (3)

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