Let P_1, P_2, ldots, P_{15} be 15 points on a | vedclass

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(D) The total number of distinct triangles that can be formed using $15$ points is given by $^{15}C_3 = \frac{15 	imes 14 	imes 13}{3 	imes 2 	ime

Vedclass · Accepted Answer

$443$

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(D) The total number of distinct triangles that can be formed using $15$ points is given by $^{15}C_3 = \frac{15 	imes 14 	imes 13}{3 	imes 2 	imes 1} = 455$.We need to exclude the cases where $i+j+k = 15$,where $1 \leq i The possible sets of $(i, j, k)$ such that $i+j+k = 15$ are:$(1, 2, 12), (1, 3, 11), (1, 4, 10), (1, 5, 9), (1, 6, 8), (2, 3, 10), (2, 4, 9), (2, 5, 8), (2, 6, 7), (3, 4, 8), (3, 5, 7), (4, 5, 6)$.Counting these,we find there are $12$ such sets.Therefore,the number of required triangles is $455 - 12 = 443$.

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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