In how many ways can $21$ English books and $19$ Hindi books be placed in a row so that no two Hindi books are together?

There are $m$ men and $2$ women participating in a chess tournament. Each participant plays $2$ games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by $84$,then the value of $m$ is

If the number of diagonals of a regular polygon of $n$ sides is $104$,then $n=$

If all the six-digit numbers $x_1 x_2 x_3 x_4 x_5 x_6$ with $0 < x_1 < x_2 < x_3 < x_4 < x_5 < x_6$ are arranged in increasing order,then the sum of the digits in the $72^{\text{th}}$ number is $............$.

Suppose that $20$ pillars of the same height have been erected along the boundary of a circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars,then the total number of beams is

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?