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

Three vertices are chosen randomly from the seven vertices of a regular $7$-sided polygon. The probability that they form the vertices of an isosceles triangle is

$m$ men and $n$ women are to be seated in a row so that no two women sit together. If $m > n$,then the number of ways in which they can be seated is

The number of diagonals in a polygon is $20$. The number of sides of the polygon is:

The number of non-congruent integer-sided triangles whose sides belong to the set $\{10, 11, 12, \ldots, 22\}$ is

If $T_n$ denotes the number of triangles which can be formed using the vertices of a regular polygon of $n$ sides and $T_{n+1}-T_{n}=21$,then $n=$