Geometric Applications of PNC

Two lines int...

Question

Two lines intersect at O. Points $A_{1}, A_{2}, . . . . A_{n}$ are taken on one of them and $B_{1}, B_{2}, . . . B_{n}$ on the other. The number of triangles that can be drawn with the help of these (2n+1) points is:

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n^{2}

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n^{3}

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n^{4}

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Solution

The correct option is C $n^{3}$
Required number of triangles $= (^{2 n} C_{3} -^{n} C_{3} -^{n} C_{3}) + (n \times n) = n^{2} (n - 1) + n^{2} = n^{3}$
Alternatively :Required number of triangles
$= (^{n} C_{2} \times^{n} C_{1}) + (^{n} C_{1} \times^{n} C_{2}) + (n \times n) = n^{3}$

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