Question

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 $i+j+k\ne 15,$ is

• A. $455$
• B. $12$
• C. $419$
• D. $443$

Answer 1

The correct option is D $443$

Total number of triangles $={}^{15}{C}_3=455$
Let $i<j<k$ so $i=1,2,3,4$ only
When $i=1,i+j+k=15$ has $5$ solutions.
$i=2,i+j+k=15$ has $4$ solutions.
$i=3,i+j+k=15$ has $2$ solutions.
$i=4,i+j+k=15$ has $1$ solutions.
Required number of triangles $=455-12=443.$