Let P_1,,P_15 be 15 points on a circle.

Question

Let $P_1,\dots,P_{15}$ be 15 points on a circle. The number of distinct triangles $P_i,P_j,P_k$ with $i+j+k
e15$ is:

Accepted Answer

Total triangles: $\binom{15}3=455$.

Count triples with $i+j+k=15$ (distinct indices from 1--15): systematically, $i=1$ gives 5 pairs, $i=2$ gives 4, $i=3$ gives 2, $i=4$ gives 1 --- total 12 such triangles.

$455-12=443$

Solutions · 1