$A$ village has $10$ players. $A$ team of $6$ players is to be formed. $5$ members are chosen out of these $10$ players and then the captain is chosen from the remaining $5$ players. The total number of ways of choosing such a team is:

The solution set of $^{10}C_{x-1} > 2 \cdot ^{10}C_x$ is

If $^{n}C_{9} = ^{n}C_{8}$,find $^{n}C_{17}$.

Consider three boxes,each containing $10$ balls labelled $1, 2, \dots, 10$. Suppose one ball is randomly drawn from each of the boxes. Denote by $n_i$ the label of the ball drawn from the $i^{th}$ box,$(i = 1, 2, 3)$. Then,the number of ways in which the balls can be chosen such that $n_1 < n_2 < n_3$ is:

If $^nC_{15} = ^nC_8$,then what is the value of $^nC_{21}$?

$A$ man has $10$ friends. In how many ways can he invite one or more of them to a party?