The number of ways in which $6$ distinct things can be distributed into $2$ boxes so that no box is empty is

There are $7$ identical white balls and $3$ identical black balls. The number of distinguishable arrangements in a row of all the balls,so that no two black balls are adjacent is

If $^nC_{r-1} = 36$,$^nC_r = 84$,and $^nC_{r+1} = 126$,then the value of $r$ is

If a committee of $10$ members is to be formed from $8$ men and $6$ women,then the number of different possible committees in which the men are in majority is

All possible two-factor products are formed from the numbers $1, 2, 3, 4, \dots, 200$. The number of products out of the total obtained which are multiples of $5$ is:

If $\frac{{}^{n+1}C_{r+1}}{{}^{n+1}C_r} = \frac{n-r+1}{m}$,then $m =$