The total number of ways of dividing $52$ cards amongst $4$ players,such that $3$ players have $17$ cards each and the fourth player has just $1$ card,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:

$n$ 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?

Three letters are dictated to three persons and an envelope is addressed to each of them. The letters are inserted into the envelopes at random so that each envelope contains exactly one letter. Find the probability that at least one letter is in its proper envelope.

There are five students $S_1, S_2, S_3, S_4$ and $S_5$ in a music class and for them there are five seats $R_1, R_2, R_3, R_4$ and $R_5$ arranged in a row,where initially the seat $R_i$ is allotted to the student $S_i$,$i = 1, 2, 3, 4, 5$. But,on the examination day,the five students are randomly allotted the five seats.
$(1)$ The probability that,on the examination day,the student $S_1$ gets the previously allotted seat $R_1$,and $NONE$ of the remaining students gets the seat previously allotted to him/her is
$(A)$ $\frac{3}{40}$ $(B)$ $\frac{1}{8}$ $(C)$ $\frac{7}{40}$ $(D)$ $\frac{1}{5}$
$(2)$ For $i = 1, 2, 3, 4$,let $T_i$ denote the event that the students $S_i$ and $S_{i+1}$ do $NOT$ sit adjacent to each other on the day of the examination. Then,the probability of the event $T_1 \cap T_2 \cap T_3 \cap T_4$ is
$(A)$ $\frac{1}{15}$ $(B)$ $\frac{1}{10}$ $(C)$ $\frac{7}{60}$ $(D)$ $\frac{1}{5}$

There are four balls of different colours and four boxes of colours same as those of the balls. The number of ways in which the balls,one in each box,could be placed such that a ball does not go to the box of its own colour is