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6 friends went to the cinema to watch a new movie. Then they realized that 2 of them are a couple and cannot be separated. How many different ways they can sit in the movie theatre?

May 24, 2020

#1
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How many seats are there in the theatre?

What is the seating plan?

Are there any more restrictions?

May 24, 2020
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6 friends went to the cinema to watch a new movie. Then they realized that 2 of them are a couple and cannot be separated. How many different ways they can sit in the movie theatre?

I'm assuming they sit in a group ... not scattered throughout the theatre.  Here's what I get:

They'll sit in 6 seats in a row.  Call the couple XX and the others A, B, C, D.

The first way is the couple first:  X X A B C D

then  X X A B D C

X X A C B D

X X A C D B

and so on.  There are 24 of these.

Move the couple over to the 2nd and 3rh seats:  A X X B C D

A X X B D C

same permutations as above.  There are 24 of these.

There are 5 positions where X X can sit.  Each of these 5 positions can seat the other 4 friends 24 ways.

So, 5 x 24 = 120 ways to seat the people.

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May 24, 2020
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I submitted my reasoning and figures.  It was fairly long.  When I clicked the Publish button, my answer disappeared.  I'm not going to go through all that again.  The bottom line I got, there are 120 ways they can sit.

Put the couple in the 1st two seats, then do the permutation of the other four people.    4 x 3 x 2 x 1 = 24

Put the couple in the 2nd two seats, then do the permutation of the other four people.                         24

Do this for all five seats the couple can sit together.  Each one has 24 ways the other four people can sit.

5 x 24 = 120

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May 24, 2020
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* I assumed that the friends would sit together in one row, and not scattered all throughout the theatre.

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Guest May 24, 2020
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Hmmm, got to thinking.  Would there be an additional 120 ways if the boy and girl switched places?  That didn't occur to me at first because I always have the girl on my left side.  Two reasons:  1. to have her near my heart.  2. that's the side my good ear is on.  So if you put the girl on the other side of the boy, there's another 120 ways and the total would be 240 ways.  The maths is getting too complicated for me; I'm going to go have some breakfast.  Happy Memorial Day, all you in the US.

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Guest May 24, 2020