Question

A team of $8$ person including ''Captain'' and ''Vice-captain'' are to be seated around a circular table, then the number of possible arrangements

• A. without any restriction is $7!$
• B. when ''Captain'' and ''Vice-captain'' should be seated opposite to each other is $5!$
• C. when there should be atleast one person between ''Captain'' and ''Vice-captain'' is $5\times 6!$
• D. when there should be exactly one person between ''Captain'' and ''Vice-captain'' is $12\times 5!$

Answer 1

Answer: (A), (C), (D)

when there should be exactly one person between ''Captain'' and ''Vice-captain'' is $12\times 5!$

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

When ''Captain'' and ''Vice-captain'' are seating opposite to each other:
Let ''Captain'' select the seat, it can be done in $1$ way (because it is circular table)
''Vice-captain'' can select the seat in $1$ way opposite to ''Captain''.
Now the remaining persons can be arranged in $6!$ ways.
$\therefore$ Required number of ways
$=1\times 1\times 6!$

When there is atleast one person between ''Captain'' and ''Vice-captain'':
Total number of ways when ''Captain'' and ''Vice-captain'' should be seated together
$=6!\times 2!$
Required number of ways
$=7!-6!\times 2!=5\times 6!$

When there is exactly one person between ''Captain'' and ''Vice-captain'':
One person can be selected in ${}^6{C}_1$ ways.
The remaining arrangements can be done in $(8-3+1-1)!=5!$ ways.
While ''Captain'' and ''Vice-captain'' can interchange among themselves.
$\therefore$ Required number of ways
$=6\times 2!\times 5!=12\times 5!$