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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!
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B
when ''Captain'' and ''Vice-captain'' should be seated opposite to each other is 5!
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C
when there should be atleast one person between ''Captain'' and ''Vice-captain'' is 5×6!
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D
when there should be exactly one person between ''Captain'' and ''Vice-captain'' is 12×5!
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Solution

The correct option is D when there should be exactly one person between ''Captain'' and ''Vice-captain'' is 12×5!
Number of ways to arrange n persons on a circular table is (n1)! ways
Total number of ways without any restriction
=(81)!=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.
Required number of ways
=1×1×6!

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!×2!
Required number of ways
=7!6!×2!=5×6!

When there is exactly one person between ''Captain'' and ''Vice-captain'':
One person can be selected in 6C1 ways.
The remaining arrangements can be done in (83+11)!=5! ways.
While ''Captain'' and ''Vice-captain'' can interchange among themselves.
Required number of ways
=6×2!×5!=12×5!

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