$n$ objects are distributed at random among $n$ persons. The number of ways in which this can be done so that at least one of them will not get any object is

If $a_n = \sum_{r=0}^n \frac{1}{^nC_r}$,then $\sum_{r=0}^n \frac{r}{^nC_r}$ equals

Number of all possible words (with or without meaning) that can be formed using all the letters of the word $CABINET$ in which neither the word $CAB$ nor the word $NET$ appear is

The number of ways of giving $20$ distinct oranges to $3$ children such that each child gets at least one orange is $............$.

The number of $4$ letter words (with or without meaning) that can be formed from the eleven letters of the word $'EXAMINATION'$ is

From a group of $7$ batsmen and $6$ bowlers,$10$ players are to be chosen for a team,which should include at least $4$ batsmen and at least $4$ bowlers. One batsman and one bowler,who are the captain and vice-captain of the team respectively,must be included. The total number of ways such a selection can be made is: