7 people, A, B, C, D, E, F and H, go to a movie and sit next to each

Question

7 people, A, B, C, D, E, F and H, go to a movie and sit next to each other in 8 adjacent seats in the front row of the theatre. In how many different arrangements will there be at least one person between A and F?

A. 5,040
B. 10,080
C. 20,160
D. 30,240
E. 35.280

A. 5,040
B. 10,080
C. 20,160
D. 30,240
E. 35.280

RupamPaul13 · Answer

Total no of possible arrangements: 8P7=40320
Now no of arrangements where at least 1 person is there between A and F = Total possible arrangements- No of arrangements where A and F are together.

No. of Arrangements where A and F are together = 2*7P6=10080

So the final answer is 40320-10080=30240(Option D)

Archit3110 · Answer

atleast = total ways - together
total ways ; 8*7*6*5*4*3*2 ; 40320
and together ; 7*6*5*4*3*2 ; 5040 ; AF can interchange so 5040*2; 10080
40320-10080 ; 30240
IMO D

Archit3110 · Answer

RupamPaul13 
see highlighted part , but we have 8 seats to make 7 such options sit ; so wont it be 8P7 ways...

RupamPaul13 · Answer

Sorry my bad, it should be 2* 7P6 as AF is taken as a single entity and they will occupy 2 seats. So 6 entity in 7 entities (seats) and AF can be arranged in 2 ways so 7P6

Lampard42 · Answer

Total arrangements - Arrangements where A and F are together
Total - 8 Seats and 7 people = 8P7 = 720*7*8 = 4320
Arrangements where A and F are together = 7 seats and 6 people = 7P6*2 (As A and F can interchange) = 720*7*2 = 1080.
So 30240 arrangements Option D

Archit3110 · Answer

Arvind42  why have you mentioned 7 seats and 6 people where as it should be 8 seats and 6 people...

Lampard42 · Answer

Archit3110  But won't A and F require 2 seats to sit. So i reduced 1 from both the seats and people.

Archit3110 · Answer

Arvind42 
when taking case of A&F together cant we take in 8P6 ways ?

Lampard42 · Answer

Archit3110 . I don't think so. Cause we join A and F together for computational purposes (still we multiply the end result by 2 as we recognize that they can interchange between themselves). So the seat count should also be reduced to that extent.

Archit3110 · Answer

Arvind42 
OK i understood ...

firas92 · Answer

Hi  Archit3110

You have negated the possibilities where A and F are sitting together but will this still not include the cases where there is an empty seat in between A and F? Whereas the question mentions there must be a person between A and F.

Please explain

Posted from my mobile device

GmatPrime · Answer

But there can be an empty seat b/w A and F in certain cases instead of a person.

How do we incorporate that into our solution ?

firas92 · Answer

Bunuel   chetan2u   VeritasKarishma

I have the same query as quoted. Can you please provide an explanation?

Posted from my mobile device

chetan2u · Answer

The question is flawed, in both wording and answer..

Wording.. 7 people, A, B, C, D, E, F and H, go to a movie and   sit next to each other in 8 adjacent seats   in the front row of the theatre.
This means all are seating next to each other, so you can have V,_,_,_,_,_,_,_ or _,_,_,_,_,_,_,V ------- so 7!*2

But ofcourse question means 
7 people, A, B, C, D, E, F and H, go to a movie and   sit on 8 adjacent seats   in the front row of the theatre.

Now, when this is what the question means..
  at least one person between A and F  implies even when there is a seat empty has to be subtracted ---- so we have to subtract (A,_,F) and 5 seats so 6!*2 from the final answer
Thus answer becomes 8!-7!*2-6!*2

But if the question was supposed to have answer as 8!-7*2!, then wordings should have been 
  at least one  person   chair  between A and F

KarishmaB · Answer

This question is a part of a series of examples in one of my blog posts to explain arrangements under constraints. There are no options given and it is not a standalone question. Hence, I have not given a lot of thought to the exact wording as long as the intent of the question is clear as in how it is different from the previous. Request you to check out the entire post to get the context and the solution in detail: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/1 ... ts-part-i/ The intent of the question is that 7 people need to sit on 8 seats in the front row. In the solution, you do have to incorporate that there might be an empty seat between A and F and the answer given is 8! – 2*7! – 2*6! (which is not in the options - 28,800)

7 people, A, B, C, D, E, F and H, go to a movie and sit next to each

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7 people, A, B, C, D, E, F and H, go to a movie and sit next to each

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