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## mathmath333 one year ago counting question

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1. mathmath333

In how many can $$4$$ men and $$4$$ women be seated around round table so that no $$2$$ men are in adjacent positions.

2. jim_thompson5910

Let's say we had 8 seats A through H |dw:1440732939919:dw|

3. jim_thompson5910

we can pull out the chairs and form a straight line |dw:1440733031604:dw|

4. jim_thompson5910

You can have it where the men go in seats A,C,E,G the women go in seats B,D,F,H

5. mathmath333

ok

6. mathmath333

u mean 4!*4! ways ?

7. jim_thompson5910

yep

8. mathmath333

the answer given is 4!*3! ways

9. anonymous

not to butt in, but here is another step

10. jim_thompson5910

go ahead, I'm missing something

11. anonymous

the way you did it you assigned a man to sit in seat A, but it could be woman so if i am not mistaken it is $$2\times 4!$$

12. anonymous

but if the answer is really $$4!3!$$ i am missing something

13. mathmath333

jim's thomson's answer is with respect to straight line, which can double count some positions on round table (circle)

14. jim_thompson5910

I guess it has something to do with this http://mathworld.wolfram.com/CircularPermutation.html

15. anonymous

oooh i see a ROUND table

16. mathmath333

yes it is linked with circular permutation

17. anonymous

the number of ways to seat 4 people in a round table is not 4! is it 3!

18. anonymous

because the table is round, we lose one place in the permutations

19. mathmath333

as stated in circular permutation formula it should be $$(4-1)!\times (4-1)!$$

20. triciaal

|dw:1440733439379:dw|

21. anonymous

i think maybe they reason like this seating four men in a round table is 3! ways, but now we have lost the circularity (if that is a word) and have four spaces in which to put the 4 women, so 4! ways there

22. anonymous

needless to say i am making this up as i go along

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