n men and n women should be arranged alternately in a row, how many ways can this be done? if the same should be done on a table, how many ways then?


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Question Number 57909 by mr W last updated on 14/Apr/19

$n m e n a n d n w o m e n s h o u l d b e a r r a n g e d$ $a l t e r n a t e l y i n a r o w, h o w m a n y w a y s$ $c a n t h i s b e d o n e ? i f t h e s a m e s h o u l d$ $b e d o n e o n a t a b l e, h o w m a n y w a y s t h e n ?$
	Answered by tanmay.chaudhury50@gmail.com last updated on 14/Apr/19

	$t r y i n g t o u n d e r s t a n d . . .$ $n m e n c a n b e a r r a n g e d i n n! w a y s$ $n o w w h e n n m e n s e a t e d . . . s e a t s a v a i l a b l e f o r$ $w o m e n a r e (n - 1) + 2 \leftarrow [t h i s 2 s e a t e x t r e m e l e f t a n d e x t r e m e r i g h t s e a t]$ $= n + 1 s e a t s$ $s o w o m e n c a n b e a r r s n g e d n + 1_{P_{n}}$ $s o a n s w e r i s n! \times (n + 1_{p_{n}})$ $= n! \times \frac{(n + 1)!}{(n + 1 - n)!}$ $= n! \times (n + 1)!$ $R o u n d t a b l e . . .$ $i n r o u n d t a b l e a t f i r s t o n e m a n s e a t t o f i x e d$ $s o n m e n c a n b e s e a t e d = (n - 1)! w a y s$ $n e x t n w o m e n c a n b e s e a t e d i n a v a i l a b l e n s e a t$ $n_{p_{n}} = \frac{n!}{(n - n)!} = n!$ $s o a n s w e r i s (n - 1)! \times n!$
	Commented by mr W last updated on 14/Apr/19

	$p l e a s e c h e c k s i r :$ $p a r t 1 :$ $n + 1 s e a t s f o r w o m e n, {t h a t}^{'} s c o r r e c t .$ $b u t n o t a l l p o s s i b i l i t i e s t o o c c u p y$ $t h e s e s e a t s a r e a l l o w e d, e . g .$ $W M W M W M W M . . . M M W M W$ $b e c a u s e m e n a n d w o m e n m u s t s i t$ $a l t e r n a t e l y .$
	Commented by tanmay.chaudhury50@gmail.com last updated on 14/Apr/19

	$t h a n k y o u s i r . . . f o r r e c t i f i c a t i o n$
	Answered by mr W last updated on 14/Apr/19

	$i n a r o w :$ $t o a r r a n g e n m e n t h e r e a r e n! w a y s .$ $t o a r r a n g e n w o m e n :$ $e i t h e r W M W M W M . . . W M W M W M \Rightarrow n! w a y s$ $o r M W M W M W . . . M W M W M W \Rightarrow n! w a y s$ $t o t a l l y 2 n! n! = 2 {(n!)}^{2} w a y s$ $o n a t a b l e :$ $t o a r r a n g e n m e n t h e r e a r e (n - 1)! w a y s .$ $t o a r r a n g e n w o m e n t h e r e a r e n! w a y s .$ $t o t a l l y n! (n - 1)!$
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