Question

If $8$ persons $A,B,C,D,E,F,G,H$ are to be seated around a circular table, then the number of possible arrangements

• A. without any restriction is $7!$
• B. when $A$ and $B$ should be seated together is $6!$
• C. when $A$ and $B$ should be seated opposite to each other is $6!$
• D. when there should be exactly two persons between $A$ and $B$ is $720$

Answer 1

Answer: (A), (C)

when $A$ and $B$ should be seated opposite to each other is $6!$

Number of ways to arrange $n$ persons in circular table is $(n-1)!$ ways
$\therefore$Total ways without any restriction is $(8-1)!=7!$

When $A$ and $B$ should be seated together
Consider $A$ and $B$ as one unit.
$\therefore$ when $A$ and $B$ should be seated together is $6!\times 2!$
$($$A$ and $B$ can interchange among themselves$)$

When $A$ and $B$ should be seated opposite to each other.
Let $A$ select the seat, it can be done in $1$ way (because it is circular table)
$B$ can select the seat in $1$ way opposite to $A$.
Now the remaining persons can be arranged in $6!$ ways.
$\therefore$Total ways are $1\times 1\times 6!$

When there should be exactly two persons between $A$ and $B$
Let the two persons be arranged in between $A$ and $B$ in${}^6{C}_2\times 2!$ ways.
The remaining $4$ places can be occupied in $4!$ ways.
Also, since $A$ and $B$ can also interchange among themselves.
$\therefore$Total ways
$={}^6{C}_2\times 2!\times 2!\times 4!=1440$