12 chairs are arranged in a row and are numbered 1 to 12.

Question

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 to 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

source:-lofoya.com

A.360

B.384

C.432

D.470

source:-lofoya.com

Bunuel · Answer

This is a poor quality question from non-GMAT source. I wouldn't worry about it at all but if still interested the solution is as follows.

First of all, I think the question means that "the chairs numbered 1 AND 8 should be occupied".

So, we have that the chairs numbered 1 AND 8 should be occupied and no two adjacent chairs must be occupied. Notice that chairs #2, 7, and 9 cannot be occupied by any of the men (because no two adjacent chairs must be occupied.).

1 - 2 -3-4-5-6- 7 - 8 - 9 -10-11-12

If the third man occupy chair #3, then the fourth man has 5 options: 5, 6, 10, 11, or 12;
If the third man occupy chair #4, then the fourth man has 4 options: 6, 10, 11, or 12;
If the third man occupy chair #5, then the fourth man has 3 options: 10, 11, or 12;
If the third man occupy chair #6, then the fourth man has 3 options: 10, 11, or 12;
If the third man occupy chair #10, then the fourth man has 1 options: 12.

Total of 5+4+3+3+1=16 cases. For each case the 4 men can be arranged in 4! ways, thus the total number of arrangements is 16*4!=384.

Answer: B.

Hope it's clear.

JOIN · Answer

Here for this question,

provided answers

If the third man occupy chair #5, then the fourth man has 3 options: 10, 11, or 12 --> fourth man has 3rd no. chair is also an option. So, total 4 options

If the third man occupy chair #6, then the fourth man has 3 options: 10, 11, or 12---> fourth man has 3rd no.,4th no. chair are also the option.So, total 5 options

If the third man occupy chair #10, then the fourth man has 1 options: 12.-->fourth man has 3rd no.,4th no.,5th no. ,6th no. chair are also the option.So, total 4 options

Please clear my doubt.

Bunuel · Answer

The options in red are covered by arrangements (given by 4!) for other options.

For example, you say that when the third man occupy chair #5, then the fourth man also has chair #3 for an option. This case is covered by arrangement (given by 4!) of scenario #1 in my solution:

1. If the third man occupy chair #3, then the fourth man has 5 options: 5, 6, 10, 11, or 12.

So, this scenario gives below cases:
   1 -2- 3 -4- 5 -6-7- 8 -9-10-11-12  .
   1 -2- 3 -4-5- 6 -7- 8 -9-10-11-12  .
   1 -2- 3 -4-5-6-7- 8 -9- 10 -11-12  .
   1 -2- 3 -4-5-6-7- 8 -9-10- 11 -12  .
   1 -2- 3 -4-5-6-7- 8 -9-10-11- 12   .

When at the end we multiply each of these case by 4!, it accounts all arrangements of 4 people on these positions.

For example, consider:    1 -2- 3 -4- 5 -6-7- 8 -9-10-11-12  . 4! here will give the arrangement when the fourth men is on the 3rd chair and the third man is on the 5th chair (as well as all other arrangements of 4 people on these chairs).

Hope it's clear.

bumpbot · Answer

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12 chairs are arranged in a row and are numbered 1 to 12.

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