Let \(P_1,P_2,\ldots,P_{15}\) be \(15\) points on a circle. The number of distinct triangles formed by points \(P_i,P_j,P_k\) such that \(i+j+k\neq 15\), is
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When a condition excludes some combinations, first count the total number of selections and then subtract the cases that violate the condition.
Step 1: Count total triangles. Total triangles from 15 points on a circle: \(\binom{15}{3}=455\).
Step 2: Subtract triangles where i + j + k = 15. Count triples \(\{i,j,k\}\) with \(1\leq i<j<k\leq15\) and \(i+j+k=15\): enumerate systematically (e.g. \(1+2+12, 1+3+11,\ldots\)) to get 12 such triples. So valid triangles \(=455-12=443\). \[ \boxed{443} \]