Let P 1 , P 2 … , 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 ≠ 15 , is :

Step-by-step solution

Given: P 1 ,   P 2 … ,   P 15 be 15 points on circle. ∴ Total number of triangles = C 3 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 ) , … … … . , ( 6 , 7 ) = 4 ways i = 3 ,   j + k = 12 ⇒ ( 4   , 8 ) , ( 5 ,   7 ) = 2 ways i