If the best and the worst paper never appear together,then in how many ways can $6$ examination papers be arranged?

In an examination hall,there are $mn$ chairs arranged in $m$ rows and $n$ columns. The number of ways in which $m$ students can be seated such that no row is vacant is:

For $n = 1, 2, 3, . . . , 50$,let $A = \{ a_n \mid a_n = \begin{cases} (-1)^{\frac{n}{2}} (\frac{n}{2}), & \text{if } n \text{ is even} \\ (-1)^{\frac{n-1}{2}} (\frac{n-1}{2}), & \text{if } n \text{ is odd} \end{cases} \}$ and $B$ be the set of all distinct elements of $A$. The number of permutations of all the elements of set $B$ such that even integers are in increasing order is:

How many permutations can be made by using all the letters of the word $MATHEMAGICA$?

The total number of injections (one-one into mappings) from the set $A = \{a_{1}, a_{2}, a_{3}, a_{4}\}$ to the set $B = \{b_{1}, b_{2}, b_{3}, b_{4}, b_{5}, b_{6}, b_{7}\}$ is:

The number of $4$ digit even numbers that can be formed using $0, 1, 2, 3, 4, 5, 6$ without repetition is