SOLUTION: Four married couples have reserved 8 seats in a row at the theatre, starting at an aisle seat. How many ways can they arrange themselves if all the women sit together and all the m

Question 33110: Four married couples have reserved 8 seats in a row at the theatre, starting at an aisle seat. How many ways can they arrange themselves if all the women sit together and all the men sit together?
Answer by longjonsilver(2297) (Show Source): You can put this solution on YOUR website!
I do not believe there is a quick "put numbers into a formula" approach to this. You need to think about it. Use a few sketches:

MMMMwwww or wwwwMMMM are the only 2 possible variations of the sets of men and women. The next thing we need to figure out is the number of ways to organise the 4 men and 4 women.

For the 4 men in MMMMwwww seating plan, there are the following number of permutations: 4! --> 24. This is gotten from the following process:

4 men. How many can sit in first seat? Answer 4.
3 remaining men. How many can sit in the second seat? Answer 3.
--> Ongoing total so far: 4*3 different permutations.
2 remaining men. How many can sit in the third seat? Answer 2.
--> Ongoing total so far: 4*3*2 different permutations.
1 remaining man. How many can sit in the fourth position? Answer 1.
--> Ongoing total so far: 4*3*2*1 different permutations.

So... 24 different ways that the 4 men can sit down.

Now the women, being 4 of them have also got 24 different ways of sitting down in their 4 seats.

For each of the 24 permutations for the men, there are 24 ways for the women to sit. So, in total, there are 24*24 --> 576 different ways for all 8 to sit in MMMMwwww.

And now, do not forget that all this is repeated for wwwwMMMM, so we have a total of 576*2 --> 1152 different permutations.

Might well be a formula, but i can never remember how to apply it to questions, so i always work it out this "from first principles" way.

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Actual working out would be something like:
MMMMwwww and wwwwMMMM

For MMMMwwww:
4 men have 4! permutations --> 24
4 women have 24 too.
Together that makes 24*24 --> 576 permutations.

wwwwMMMM has the same number.
Hence total is 576+576 --> 1152 permutations

jon.

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