There are $5$ points $P_1, P_2, P_3, P_4, P_5$ on the side $AB$,excluding $A$ and $B$,of a triangle $ABC$. Similarly,there are $6$ points $P_6, P_7, \ldots, P_{11}$ on the side $BC$ and $7$ points $P_{12}, P_{13}, \ldots, P_{18}$ on the side $CA$ of the triangle. The number of triangles that can be formed using the points $P_1, P_2, \ldots, P_{18}$ as vertices is:

Let $n > 2$ be an integer. Suppose that there are $n$ Metro stations in a city located along a circular path. Each pair of stations is connected by a straight track. Further,each pair of nearest stations is connected by a blue line,whereas all remaining pairs of stations are connected by a red line. If the number of red lines is $99$ times the number of blue lines,then the value of $n$ is:

If four points are taken on each of three parallel lines in a plane,then the maximum number of triangles formed with these points is

Given six line segments of lengths $2, 3, 4, 5, 6, 7$ units,the number of triangles that can be formed by these lines is

Let $n \geq 2$ be an integer. Take $n$ distinct points on a circle and join each pair of points by a line segment. Colour the line segment joining every pair of adjacent points blue and the rest red. If the number of red and blue line segments are equal,then the value of $n$ is

$A$ tea party is arranged for $16$ people at a long table with $8$ chairs on each side. $4$ specific people wish to sit on one particular side and $2$ on the other side. In how many ways can they be seated?