$A$ polygon has $54$ diagonals. The number of sides of this polygon is

Some couples participated in a mixed doubles badminton tournament. If the number of matches played,such that no couple played in a match,is $840$,then the total number of persons who participated in the tournament is $........$.

In a row,$6$ boys and $6$ girls are arranged at random. Find the probability that all $6$ girls are together.

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

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