In how many ways can $5$ prizes be distributed among four students when every student can take one or more prizes?

The number of integers greater than $6000$ that can be formed using the digits $3, 5, 6, 7,$ and $8$ without repetition is:

If $(n+1)! = 6[(n-1)!]$,find $n$.

$A$ box contains $2$ white balls,$3$ black balls and $4$ red balls. In how many ways can $3$ balls be drawn from the box,if at least one black ball is to be included in the draw?

If the letters of the word $KRISNA$ are arranged in all possible ways and these words are written out as in a dictionary,then the rank of the word $KRISNA$ is

The number of ways of choosing $10$ objects out of $31$ objects,of which $10$ are identical and the remaining $21$ are distinct,is