Let $P_1,P_2,P_3,\mathrm{....},P_{15}$ be 15 points on a circle. The number of triangles (distinct) formed by the points $P_i,P_j,P_k$ such that $i+j+k\ne 15$ is

Total number of triangles formed  using 15 concyclic points= ${ }^{15}{C}_3$

Number  of triangles $P_i,P_j,P_k$ such that $i+j+k\ne 15$ is

$\mathrm{Total} \mathrm{number} \mathrm{of} \mathrm{triangles}-(i+j+k=15){=}^{15}{C}_3-(5+4+2+1)=443$
$i=1\to (j,k)=(2,12)(3,11)(4,10)(5,9)(6,8)\to 5$
$\begin{array}{l}i=2\to (3,10)(4,9)(5,8)(6,7)\to 4\\ i=3\to (4,8)(5,7)\to 2\\ i=4\to (5,6)\to 1\end{array}$