There are $7$ greeting cards, each of a different colour, and $7$ envelopes of the same $7$ colours as the cards. The number of ways in which the cards can be put in envelopes, so that exactly $4$ of the cards go into envelopes of the respective colour, 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
$(A)$ $\frac{1}{15}$ $(B)$ $\frac{1}{10}$ $(C)$ $\frac{7}{60}$ $(D)$ $\frac{1}{5}$

There are $4$ distinct colored balls and $4$ boxes of the same colors as the balls. In how many ways can the balls be placed in the boxes such that no ball goes into a box of its own color?

If $4$ letters are placed randomly into $4$ envelopes,what is the probability that none of the letters are placed in their correct envelopes?

The number of ways in which $9$ persons can be divided into three equal groups is

Three letters are to be sent to different persons,and addresses on the three envelopes are also written. Without looking at the addresses,the probability that all the letters go into the right envelopes is equal to