In an examination,$5$ students have been allotted their seats as per their roll numbers. The number of ways,in which none of the students sits on the allotted seat,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:

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

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

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

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