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.

$A$ person writes letters to $6$ friends and addresses the corresponding envelopes. In how many ways can the letters be placed in the envelopes so that at least two of them are in the wrong envelopes?
Notation : $D_n = n! \left( \sum_{i=0}^n \frac{(-1)^i}{i!} \right)$

There are $n$ letters and $n$ addressed envelopes. The probability that all the letters are not kept in the right envelope is:

The number of ways in which $9$ persons can be divided into three equal groups is

Let the set $S = \{2, 4, 8, 16, \ldots, 512\}$ be partitioned into $3$ sets $A, B, C$ with an equal number of elements such that $A \cup B \cup C = S$ and $A \cap B = B \cap C = A \cap C = \phi$. The number of such possible partitions of $S$ is equal to:

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