Three letters,to each of which corresponds an envelope,are placed in the envelopes at random. The probability that all the letters are not placed in the right envelopes is

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

$n$ distinct items $1, 2, 3, \dots, n$ are arranged in $n$ distinct positions $1, 2, 3, \dots, n$. What is the probability that at least three items are in their correct positions?

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:

$150$ students took admission. In how many ways can they be divided into three equal sections $A, B,$ and $C$?

In how many ways can a pack of $52$ cards be divided into four equal groups?