Six '$X$'s have to be placed in the squares of the figure such that each row contains at least one '$X$'. In how many different ways can this be done?

Seven white balls and three black balls are randomly placed in a row. The probability that no two black balls are placed adjacently equals

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.

There are $n$ distinct points on a circle. If the number of pentagons that can be formed using these points as vertices is equal to the number of triangles that can be formed,then the value of $n$ is:

Ten points lie in a plane such that no three of them are collinear. The number of lines passing through exactly two of these points and dividing the plane into two regions,each containing four of the remaining points,is

If three of the six vertices of a regular hexagon are chosen at random,then the probability that the triangle formed with these chosen vertices is equilateral is