Permutations and combinations

Question

Six friends go to watch a movie. They are supposed to occupy seat numbers 51 to 56. However one of them falls sick and returns home. In how many different ways can the 5 people sit?

Hello people!

I have come across this question in one of the gmatclub theory threads by Veritas and Bunuel. The problem is that there isn't any solution for the above mentioned question. Could anyone please help me with the solution for the same problem?

Thanks in advance!

TopTierMBA · Answer

Problem is how 5 people can occupy 6 seats in a row.
Well 1st person can occupy a seat in 6 ways, the 2nd person in 5 ways, and so on.
So, total no. of ways in which 5 people can occupy 6 seats = 6 x 5 x 4 x 3 x 2 = 720.

Else,
5 people can select 5 seats from a row of 6 seats in 6c5 ways viz. 6 ways
These 5 people can arrange themselves in 5! = 120 ways.
So total no. of ways in which 5 people can occupy 6 seats = 6 x 120 = 720.

Hope this helps.

Cheers!

EMPOWERgmatRichC · Answer

Hi INSEADIESE,

Unless there are any additional 'restrictions' to how the 5 people can sit in the 6 seats, the fact that the 6th person is not there does NOT change the end result. The one 'empty' seat just ends up as an entity that you have to account for (re: any of the 6 seats can be the empty seat) - meaning that we are dealing with a Permutation in which we are placing 5 people and 1 blank seat (re: 6 entities) and that calculation would be (6)(5)(4)(3)(2)(1) = 720 possible arrangements.

However, if there are restrictions included (for example, the "end" seats - 51 and 56 - both must be filled or the 5 people must fill 5 consecutive seats), then the result would change. 
 
GMAT assassins aren't born, they're made,
Rich

goaltop30mba · Answer

Thank you <img src="{SMILIES_PATH}/1f642.png" class="forumSmileyItem" width="20" height="20" alt=":)" title="Smile" />

goaltop30mba · Answer

Understood Thank you

ScottTargetTestPrep · Answer

Let the 5 people who are present at the movie theater be A, B, C, D and E and let _ denote the empty seat. So we could have:

_ABCDE, A_BCDE, AB_CDE, ABC_DE, ABCD_E and ABCDE_

However, for each of these 6 seating arrangements, there are 5! to arrange the 5 people while keeping the empty seat as where it is. For example, for A_BCDE (i.e., the second seat is empty), we could have C_DABE, D_EBAC, etc.

Therefore, there are a total of 6 x 5! = 6 x 120 = 720 seating arrangements.

Answer: 720

goaltop30mba · Answer

thank you <img src="{SMILIES_PATH}/1f642.png" class="forumSmileyItem" width="20" height="20" alt=":)" title="Smile" />

goaltop30mba · Answer

Hello People,

here is another pnc question that i have come across in the math theory section available here on gmatclub. Again, there isn't any solution for the question. Could anyone please help me with the solution for the same problem?

here it goes :-

6 people go to a movie and sit next to each other in 6 adjacent seats in the front row of the theatre. If A cannot sit to the right of F, how many different arrangements are possible?

EMPOWERgmatRichC · Answer

Hi INSEADIESE,

Since we're placing 6 people in a ROW, we're clearly dealing with a Permutation - and there would be (6)(5)(4)(3)(2)(1) = 720 possible arrangements.

Regardless of the number of people involved, the probability that one person would sit anywhere to the left of, or anywhere to the right of, another person is 50%.

For example, if there were 2 people, then possible arrangements would be:
AB
BA

Half the time, A sits to the left of B and half the time, A sits to the right of B

if there were 3 people, then possible arrangements would be:
ABC
ACB
BAC
BCA
CAB
CBA

Half the time, A sits to the left of B and half the time, A sits to the right of B

Etc.

With 6 people, if A CANNOT sit to the right of F, then that would eliminate HALF of the possible arrangements, and there would be 720/2 = 360 possible arrangements.

GMAT assassins aren't born, they're made,
Rich

goaltop30mba · Answer

Understood Thank You

ScottTargetTestPrep · Answer

Of course.

Permutations and combinations

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