In how many ways can $52$ cards be distributed equally among four players?

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?

There are $4$ envelopes with addresses and $4$ corresponding letters. The probability that no letter goes into its correct envelope is:

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

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?

There are $8$ different coloured balls and $8$ bags having the same colours as that of the balls. If one ball is placed at random in each one of the bags,then the probability that $5$ of the balls are placed in the respective coloured bags,is