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$12$ chairs are arranged in a row and are numbered $1$ to $12$. $4$ men have to be seated in these chairs so that the chairs numbered $1$ ans$8$ should be occupied and no two men occupy adjacent chairs.

Find the number of ways the task can be done.









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Option(B) is correct

Given there are $12$ numbered chairs, such that chairs numbered $1$ and $8$ should be occupied.

$\_X\_ , 2,3,4,5,6,7,\_X\_,9,10,11,12.$

The various combinations of chairs that ensure that no two men are sitting together are listed.

$(1,3,5,\_\_)$. The fourth chair can be $5,6,10,11$ or $12$, hence $5$ ways.

$(1,4,8,\_\_)$, The fourth chair can be $6,10,11$ or $12$ hence $4$ ways.

$(1,5,8,\_\_)$, the fourth chair can be $10,11$ or $12$ hence $3$ ways.

$(1,6,8,\_\_)$, the fourth chair can be $10,11$ or $12$ hence $3$ ways.

$(1,8,10,12)$ is also one of the combinations.

Hence, $16$ such combinations exist.

In the case of each these combinations, we can make the four men inter arrange in $4!$ ways.

Hence, the required result,

$= 16× 4! = \textbf{384}$

Edit: After input from Gail Mitchell question hase been updated from  '$1$ to $8$' to '$1$ and $8$'.

Edit: for an alternative solution, check comment from Raj.

(10) Comment(s)


Please frame questions correctly, otherwise, it will kill user's confidence ,interest. ie; Exactness, appropriate words. increase difficulty in L2, L3 levels.Let users able to understand easily in Basic level


If position 1 and 8 are occupied then when

  • Both the remaining men sitting left of 8......then the following positions are possible for the two men (3,5);(3,6) and (4,6) 3 arrangement possible
  • Both the remaining men sitting right of 8.... only position (10,12) is possible for them to 1 arrangement possible.
  • One remaining man sitting left of 8 and the other sit right of 8 in this case left man can occupy any of (3,4,5,6) position and right man can occupy any of (10,11,12) $4*3=12$ arrangement is possible.

Thus, in total $3+1+12=16$ arrangement are possible.

and 4 men can be arranged in 4! ways, therefore, answer $=16*4!$

Sushma Saroj

I also didn't understand the question and it's answer.plz help me


Is this correct?

I've enumerated 768 arrangements.


coudnt understand the question and its explanation!Sad

Shaswat Khamari

It should be "chairs 1 and 8 must be occupied."

Shamsa Kanwal

there should be 8 people shouldn't


Even after that I think the answer is wrong..

Chirag Khimani

ya I think it is a mistake...

Gail Mitchell

Shouldn't the wording be "the chairs 1 AND 8" must be occupied?