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

Five letters are placed at random in five addressed envelopes. The probability that all the letters are not dispatched in the respective right envelopes is

Let $A = \{x_1, x_2, x_3, x_4\}$ and $B = \{y_1, y_2, y_3, y_4\}$. $A$ function $f: A \to B$ is defined. The number of one-one functions such that $f(x_i) \neq y_i$ for $i = 1, 2, 3, 4$ is equal to:

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 $7$ greeting cards, each of a different colour, and $7$ envelopes of the same $7$ colours as the cards. The number of ways in which the cards can be put in envelopes, so that exactly $4$ of the cards go into envelopes of the respective colour, is:

The number of ways of distributing $3$ dozen fruits (no two fruits are identical) to $9$ persons such that each gets the same number of fruits is