If in a regular polygon the number of diagonals is $54$,then the number of sides of this polygon 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

Given $6$ different letters of an alphabet. Words with $4$ letters are formed from these given letters. The number of words which have at least one letter repeated and no two same letters are together,is:

The value of $^{15}C_3 + ^{15}C_{13}$ is

If the number of $5$-element subsets of the set $A = \{a_1, a_2, \dots, a_{20}\}$ of $20$ distinct elements is $k$ times the number of $5$-element subsets containing $a_4$,then $k$ is

The number of diagonals in a polygon of $m$ sides is