If $5$ letters are to be placed in $5$ addressed envelopes,then the probability that at least one letter is placed in the wrongly addressed envelope is

If there are $5$ letters written to $5$ different people and $5$ envelopes addressed to them,then the number of ways in which these letters can be arranged so that no letter goes into its corresponding envelope 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:

There are $4$ letters and $4$ envelopes. If the letters are placed into the envelopes at random,find the probability that all letters are placed in the wrong envelopes.

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

Six cards and six envelopes are numbered $1, 2, 3, 4, 5, 6$. Cards are to be placed in envelopes such that each envelope contains exactly one card,no card is placed in the envelope bearing the same number,and the card numbered $1$ is always placed in the envelope numbered $2$. The number of ways this can be done is: