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We are seating 5 married couples around a table (the seats are identical). Let {M1, M2, M3, M4, M5} be the set of men and let {W1, W2, W3, W4, W5} be the set of their wives.

In how many cases the man 1 will be seated next to his wife and the man 3 will not seat next to his wife?

My idea: We are putting M3 anywhere and than choosing the places next to him. It can't be W3 so we have (7 for one place) and (6 for other place), we can change the order so it's *2. There are 6 places left so we have 6!(we are counting M1 and W1 as one seat)

Solution is: 2*7*6*6! or 2*7*6*6!*2

Is it ok? (which one)

2 Answers 2

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Seat M1 anywhere.

W1 can go in two places next to M1.

M3 now either goes next to M1, or next to W1, or somewhere else. In the first two cases, there are 6 seats left for W3, and in the other 6 cases there are 5 seats left for W3.

Once these are seated, the last 6 can go anywhere

So: 2 * (2*6+6*5)*6! = 2 * 7 * 6 * 6!

So that is almost the same as your answer, but you have an extra 2 ... and I am not sure where you get the extra 2 from ..

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    I have extra *2 from putting second and third in reverse order. But I think your is better. – 2017-02-05
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    because I think there is no difference: m1 m3 m2 and m2 m3 m1 if there is no difference it should be without extra 2 – 2017-02-05
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    @sswwqqaa Right: if you have A placed and then place B, you get A B X X X X X X X X but also A X X X X X X X X B, and the latter is the same as B A X X X X X X X X, i.e. where A and B are switched. – 2017-02-05
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    ok, now I think I understand, thanks – 2017-02-05
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I think that will be great, but not sure.

We are seating 1st man (We are using round table, so we have not to include order) 1st man have to sit with his wife so - 1*2

Next we are taking 3st man, he has 8 seats to take so 1*2*8

Wife of 3st man cant seat next to him so 6 places are free 1*2*8*6

And we have 6 people to 6 places so 6! will be best.

So final answer is 1*2*8*6*6!

I'm not sure that is right but this is my way of thinking.

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    not quite ... if you place M3 not right next to either M1 or W1, then W3 has only 5 places left, not 6. – 2017-02-05