Permutation-Combination
Aptitude

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Q.

$12$ 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.

 A.

$360$

 B.

$384$

 C.

$432$

 D.

$470$

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Solution:
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)


Satish
 ()

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



Raj
 ()

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).....ie 3 arrangement possible
  • Both the remaining men sitting right of 8.... only position (10,12) is possible for them to occupy.....ie 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) position.....ie $4*3=12$ 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



Martin
 ()

Is this correct?

I've enumerated 768 arrangements.



Abi
 ()

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



Sagar
 ()

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?