If there are $4$ letters and $4$ envelopes,in how many ways can all the letters be placed in the wrong envelopes?

Consider a square matrix of order $5$ such that $a_{ij} = 0$ for all $i + j = 6$,where $a_{ij} \in \{0, 1\}$ for all $i, j$. In each row as well as in each column,there is only one non-zero element. Then,the number of such matrices is:

Eight persons are to be transported from city $A$ to city $B$ in three cars of different makes. If each car can accommodate at most three persons,then the number of ways,in which they can be transported,is $...........$.

The number of ways of dividing $200$ dissimilar things into $10$ groups each containing $20$ elements is

$A$ person has three different bags and four different books. The number of ways,in which he can put these books in the bags so that no bag is empty,is:

There are four balls of different colours and four boxes of colours same as those of the balls. The number of ways in which the balls,one in each box,could be placed such that a ball does not go to the box of its own colour is