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-

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 \dots

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 \dots 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 \boldsymbol n! \times (\boldsymbol n + 1_{\boldsymbol p_{\boldsymbol n}})

= n! \times \frac{(n + 1)!}{(n + 1 - n)!}

= n! \times (n + 1)!

R o u n d t a b l e \dots

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 \dots 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 \dots 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 \dots W M W M W M \Rightarrow n! w a y s

o r M W M W M W \dots 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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