In how many ways can $3$ letters be posted in $4$ letter boxes,if all the letters are not posted in the same letter box?

The number of seven-digit positive integers formed using the digits $1, 2, 3,$ and $4$ only,such that the sum of the digits is equal to $12$,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:

In how many ways can the letters of the word $ASSASSIN$ be arranged so that no two $S$s occur together?

The number of $5$-digit numbers that are not divisible by $5$ and consist of different odd digits is:

If ${}^n C_{r-1}=36$, ${}^n C_r=84$, and ${}^n C_{r+1}=126$, then the value of ${}^n C_8$ is