4 couples are to take a photograph with a newly wedded couple in a wedding party.In how many ways can this be done if: i)the celebrated couple must stand in the middle ii)each couple must stand next to each other iii)the celebrated couple must not stand next to each other


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Question Number 34877 by NECx last updated on 12/May/18

$4 c o u p l e s a r e t o t a k e a p h o t o g r a p h$ $w i t h a n e w l y w e d d e d c o u p l e i n a$ $w e d d i n g p a r t y . I n h o w m a n y w a y s$ $c a n t h i s b e d o n e i f :$ $i) t h e c e l e b r a t e d c o u p l e m u s t s t a n d$ $i n t h e m i d d l e$ $i i) e a c h c o u p l e m u s t s t a n d n e x t t o$ $e a c h o t h e r$ $i i i) t h e c e l e b r a t e d c o u p l e m u s t n o t$ $s t a n d n e x t t o e a c h o t h e r$
Commented by NECx last updated on 12/May/18

$I^{'} m s o s o r r y I m a d e t y p o e r r o r . I t s$ $m e a n t t o b e 4 c o u p l e s n o t A c o u p l e .$ $I t h a s b e e n e d i t d d .$
	Answered by tanmay.chaudhury50@gmail.com last updated on 12/May/18

	$l e t n e w c e l e b r a t e d c o u p l e H_{n}, W_{n}$ $o l d c o u p l e H_{o}, W_{o}$ $1) H_{o}, H_{n}, W_{n}, W_{o}$ $H_{o}, W_{n}, H_{n}, W_{o}$ $W_{o}, W_{n}, H_{n}, H_{o}$ $W_{o}, H_{n}, W_{n}, H_{o}$ $f o u r p l a c e t o b e f i l l e d . . . t w o e x t r e m e b y 2 w a y s$ $a n d m i d d l e t w o p l a c e b y t w o n e w c o u p l e$ $s o 2 \times 2 = 4 w a y s$ $2) H_{n} W_{n} H_{o} W_{o}$ $H_{n} W_{n} W_{o} H_{o}$ $W_{n} H_{n} H_{o} W_{o}$ $W_{n} H_{n} W_{o} H_{o}$ $H_{o} W_{o} H_{n} W_{n}$ $H_{o} W_{o} W_{n} H_{n}$ $W_{o} H_{o} H_{n} W_{n}$ $W_{o} H_{o} W_{n} H_{n}$ $s o 8 w a y s$ $3) t h e e x t r e m e t w o e n d p l a c e c a n b e f i l l e d b y e i t h e r$ $n e w h u s b a n d o r n e w w i f e s o 2 w a y s$ $m i d d l e t w o p l a c e b y 2 w a y s s o t o t a l 2 \times 2 = 4 w a y s$
	Answered by tanmay.chaudhury50@gmail.com last updated on 12/May/18

	$1) t o t a l m e m b e r i n t h e p a r t y = 4 \times 2 + 2 = 10$ $1 2 3 4 5 6 7 8 9 10$ $t h e r e a r e t e n s e a t t o b e o c c u p i e d b y p a r t y m e m b d r$ $n e w c o u p l e t o o c c u p y s e a t n u m b e r 5 a n d 6 s$ $r e m a i n i n g e i g h t s e a t t o b e o c c u p i e d b y o t h e r$ $f o u r c o u p l e$ $s o 2! \times 8!$ $2) i m a g i n e e a c h c o u p l e a r e a t t a c h e d t o g e t h e r$ $s o o n e c o u p l e m a y b e i m a g i n e d o n e u n i t$ $s o r e q u i r e d p e r m u t a t i o n i s 5! \times 2!$ $3) l e t c e l e b r a t e d c o u p l e a r e f a b i q u i c k e d w i t h$ $e a c h o t h e r s o t h a t t h e y c a n n o t b e s e p a r a t e d$ $t h e p e r m u t a t i o n s o t b a t n e w c o u p l e a l w a y s$ $t o g e t h e r i s (4 \times 2 + 1)! \times 2!$ $= 9! \times 2!$ $s o r e q u i r e d p e r m u t a t i o n w h e n n e w c o u p l e$ $n e v e r t o g e t h e r i s 10! - 9! \times 2!$
	Commented by NECx last updated on 12/May/18

	$T h a n k y o u s o m u c h$
	Commented by NECx last updated on 13/May/18

	$i t s r e a l l y e x p l a n a t o r y$
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