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 $..........$.

The number of ways in which $4$ letters can be put into $4$ addressed envelopes such that no letter goes into the envelope meant for it is:

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

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:

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
$(A)$ $\frac{1}{15}$ $(B)$ $\frac{1}{10}$ $(C)$ $\frac{7}{60}$ $(D)$ $\frac{1}{5}$

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.