Question

$12$ persons are to be seated at a square table, three on each side. $2$ persons wish to sit on the north side and two wish to sit on the east side. One other person insists on occupying the middle seat(which may be on any side). Find the number of ways they can be seated.

Answer 1

Question will be solved with respect to the person who insists to sit in the middle.

Case (i) When he sits on West or South side

He will have 2 options and then group of 2 people who wish to sit on North and East side respectively can be arranged in the available 3 places in their respective direction.

Finally 7 people will be left which can be arranged in the 7 possible ways,

ie., $2\times {}_2^3\text{P}\times {}_2^3\text{P}\times 7!$

$=2\times 6\times 6\times 7!$

$=72\times 7!=9\times 8\times 7!=9\times 8!$

Case (ii) When he sits on North or East side

He will have 2 options to sit on either North or East side and then one of the group of 2 persons, who wish to sit on North and East side respectively will be 3 places to sit while the other group will have 2 options.

Finally 7 people will be left which can be arranged in the 7 possible ways,

ie., $2\times {}_2^3\text{P}\times {}_2^2\text{P}\times 7!$

$=2\times 6\times 2\times 7!$

$=24\times 7!=3\times 8\times 7!=3\times 8!$

Total Ways $=9\times 8!+3\times 8!$

$=12\times 8!$

$=483840$