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?

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

The number of ways of dividing $200$ dissimilar things into $10$ groups each containing $20$ elements 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 set $S = \{1, 2, 3, \dots, 12\}$ is to be partitioned into three sets $A, B, C$ of equal size such that $A \cup B \cup C = S$ and $A \cap B = B \cap C = C \cap A = \emptyset$. The number of ways to partition $S$ is:

If there are $6$ letters and $6$ corresponding envelopes,in how many ways can all the letters be placed in the wrong envelopes?