Question
Two straight line intersect at a point $$O.$$ Points $${A_1},{A_2},.....,{A_n}\,$$ are taken on one line and points $${B_1},{B_2},.....,{B_n}\,$$ on the other. If the point $$O$$ is not to be used, the number of triangles that can be drawn using these points as vertices, is
A.
$$n\left( {n - 1} \right)$$
B.
$$n{\left( {n - 1} \right)^2}$$
C.
$$n^2 {\left( {n - 1} \right)}$$
D.
$$n^2 {\left( {n - 1} \right)^2}$$
Answer :
$$n^2 {\left( {n - 1} \right)}$$
Solution :
No. of triangles $$ = {\,^{2n}}{C_3} - {\,^n}{C_3} - {\,^n}{C_3}$$
$$\eqalign{
& = \frac{{2n\left( {2n - 1} \right)\left( {2n - 2} \right)}}{6} - \frac{{2n\left( {n - 1} \right)\left( {n - 2} \right)}}{6} \cr
& = \frac{1}{3}n\left( {n - 1} \right)\left( {3n} \right) = {n^2}\left( {n - 1} \right). \cr} $$