If $x, y$ and $r$ are positive integers, then $^x{C_r} + ^x{C_{r-1}} \cdot ^y{C_1} + ^x{C_{r-2}} \cdot ^y{C_2} + \dots + ^y{C_r} = $

If there are $6$ periods in each working day of a school,in how many ways can one arrange $5$ subjects such that each subject is allowed at least $1$ period?

If $C(n, 8) = C(n, 6)$,find $C(n, 2)$.

The number of $5$ digit telephone numbers having at least one of their digits repeated is

If $n = ^mC_2,$ then the value of $^nC_2$ is given by

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: