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

The set $S = \{1, 2, 3, \dots, 12\}$ is partitioned into three sets $A, B, C$ of equal size such that $A \cup B \cup C = S$ and $A \cap B = B \cap C = C \cap A = \phi$. In how many ways can $S$ be partitioned?

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

Let $\alpha = \frac{(4!)!}{(4!)^{3!}}$ and $\beta = \frac{(5!)!}{(5!)^{4!}}$. Then:

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

$A$ 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: