The number of triangles that can be formed by choosing the vertices from a set of $12$ points,$7$ of which lie on the same straight line,is

The number of ways in which $3$ boys and $2$ girls can sit on a bench so that no two boys are adjacent is . . . . . .

There is a rectangular sheet of dimension $(2m - 1) \times (2n - 1)$,(where $m > 0, n > 0$). It has been divided into squares of unit area by drawing lines perpendicular to the sides. Find the number of rectangles having sides of odd unit length.

$15$ persons are sitting around a circular table. The number of ways of selecting three persons at a time from them,such that the selected three do not sit together at one place is

Out of $30$ points in a plane,$8$ of them are collinear. The number of straight lines that can be formed by joining these points is:

Three of the six vertices of a regular hexagon are chosen at random. The probability that the triangle formed by these three vertices is equilateral is equal to