Let ABC be a triangle. Consider four points on the side AB, five points on the side BC, and four points on the side AC. None of these points is a vertex of triangle ABC. [2026]

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Solution

Q.
Let ABC be a triangle. Consider four points $p_{1}, p_{2}, p_{3}, p_{4}$ on the side AB, five points $p_{5}, p_{6}, p_{7}, p_{8}, p_{9}$ on the side BC, and four points $p_{10}, p_{11}, p_{12}, p_{13}$ on the side AC. None of these points is a vertex of triangle ABC. Then the total number of pentagons that can be formed by taking all the vertices from the points $p_{1}, p_{2}, \dots, p_{13}$ is _________ . [2026]

Ans.
(660)

$Case 1 :$

$2 from A B, 2 from B C, 1 from A C$

$(\binom{4}{2}) . (\binom{5}{2}) . (\binom{4}{1}) = 6 \cdot 10 \cdot 4 = 240$

$Case 2:$

$2 from A B, 1 from B C, 2 from A C$

$(\binom{4}{2}) . (\binom{5}{1}) . (\binom{4}{2}) = 6 \cdot 5 \cdot 6 = 180$

$Case 3:$

$1 from A B, 2 from B C, 2 from A C$

$(\binom{4}{1}) . (\binom{5}{2}) . (\binom{4}{2}) = 4 \cdot 10 \cdot 6 = 240$

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