If $4$ letters are placed randomly into $4$ envelopes,what is the probability that none of the letters are placed in their correct envelopes?

There are $4$ balls of different colors and $4$ boxes of the same colors. In how many ways can the $4$ balls be placed in the boxes,one in each box,such that no ball goes into a box of its own color?

The number of ways of dividing $52$ cards amongst four players equally is

In an examination,$5$ students have been allotted their seats as per their roll numbers. The number of ways,in which none of the students sits on the allotted seat,is $..........$.

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:

Three letters are placed at random into three envelopes addressed to three different people. The probability that all letters are placed in the correct envelopes is ..........