Let P _ 1 , P _ 2 , P _ 15 be 15 points on a circle. The number of...

Given:${P}_{1}, {P}_{2} \ldots , {P}_{15}$ be $15$ points on circle.

$∴$Total number of triangles$= C3 15$

When $i + j + k = 15$ then possible cases are

$i = 1, j + k = 14 ⇒ ( 2, 12 ) , ( 3, 11 ) , ( 4, 10 ) , ( 5, 9 ) , ( 6, 8 ) = 5$ ways

$i = 2, j + k = 13 ⇒ ( 3, 10 ) , \ldots \ldots \ldots . , ( 6, 7 ) = 4$ ways

$i = 3, j + k = 12 ⇒ ( 4, 8 ) , ( 5, 7 ) = 2$ ways

$i = 4, j + k = 11 ⇒ ( 5, 6 ) = 1$ way

Hence, there are total $12$ ways for $i + j + k = 15$.

$∴$ The number of possible triangles using the vertices ${P}_{i}, {P}_{j}, {P}_{k}$ such that $i + j + k \ne 15$ is equal to $C3 15 - 12 = 455 - 12 = 443$.