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A group of 5 men and 5 women stand in line to have their photo taken.

How many ways can they stand in line if no two men and no two women stand together?

My method: _M_M_M_M_M_

Male * Female = 5P5 * 6P5 = 86400

Correct Answer = 5! * 5! * 2 = 28800

I don't understand why I got it wrong, can anyone help please?

user9856
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  • Why do you have 6 $_$s? – Peter Woolfitt Jun 27 '14 at 23:18
  • They are the spaces for the women, I used the logic of this question http://math.stackexchange.com/questions/12587/how-many-ways-are-there-for-8-men-and-5-women-to-stand-in-a-line-so-that-no-two – user9856 Jun 27 '14 at 23:19
  • right, but you have 6 of them and only 5 women – Peter Woolfitt Jun 27 '14 at 23:21
  • Yes, so that's why I put 6P5, arranging 5 women in 6 spaces. Is that correct? – user9856 Jun 27 '14 at 23:22
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    No it is not. Here neither men nor women can be neighbours. In the other question, men could be neighbours (and, given the numbers, some had to be). – André Nicolas Jun 27 '14 at 23:23
  • Sorry I dont understand that... Isn't my method going to separate all of them so they are arranged alternately? – user9856 Jun 27 '14 at 23:24
  • @André OP's reasoning makes sure both men and women are not neighbors. (It's invalid for another reason...) – anon Jun 27 '14 at 23:25
  • Definitely not. If you assign a woman to the leftmost $3$ spaces, and to the rightmost $2$, a couple of men around the middle will be next to each other. – André Nicolas Jun 27 '14 at 23:25
  • Oh... I see why my method is wrong now. Can you explain how the solution works? – user9856 Jun 27 '14 at 23:26
  • Order the five women, order the five men, then decide whether the seats start with a man or women and interlace them. (5! x 5! x 2) @Andre Ah, duh. – anon Jun 27 '14 at 23:28
  • Oh I didn't know that you could do that... I think I get some of it now... Is there a more 'graphical' solution? :) – user9856 Jun 27 '14 at 23:30
  • @blue: Thank you for informing me of a way to solve the problem. I might have floundered helplessly otherwise. For days. – André Nicolas Jun 27 '14 at 23:30
  • @André Sorry, I should have phrased it in the form of a hint to the OP, I know. (e.g. "Count the ways depending on whether a man or woman sits first.") – anon Jun 27 '14 at 23:32

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If you distiguish only by man and woman, then they can be arranged: $$m-w-m-w-m-w-m-w-m-w$$ and $$w-m-w-m-w-m-w-m-w-m$$

If you distinguish the womans, there are $5!$ possibilities. The same for men.

callculus42
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