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Question Number 113486 by mr W last updated on 13/Sep/20

Commented by mr W last updated on 13/Sep/20

$R e p o s t Q 112509$
	Answered by mr W last updated on 13/Sep/20

	$40 c h a i r s b u i l d 20 p a i r s w i t h o p p o s i t e$ $c h a i r s .$ $f o r 17 b o y s t o t a k e t h e 40 c h a i r s o n$ $t h e t a b l e t h e r e a r e t o t a l l y P_{17 - 1}^{40 - 1} w a y s .$ $t o s e l e c t t w o b o y s w h o s i t o p p o s i t e$ $t h e r e a r e C_{2}^{17} w a y s .$ $t o s e l e c t 15 f r o m 19 c h a i r p a i r s t h e r e$ $a r e C_{15}^{19} w a y s . t o a r r a n g e t h e 15 b o y s$ $i n t h e s e c h a i r s t h e r e a r e 15! \times 2^{15}$ $w a y s .$ $t o t a l l y C_{2}^{17} \times C_{15}^{19} \times 15! \times 2^{15} .$ $p = \frac{C_{2}^{17} \times C_{15}^{19} \times 15! \times 2^{15}}{P_{16}^{39}} = 2 . 8627 %$
	Commented by I want to learn more last updated on 13/Sep/20

	$Thanks sir, i really appreciate$
	Commented by I want to learn more last updated on 14/Sep/20

	$Sir, i have the questions .$ $(1) I thought for circular arrangement, for this case, we should select$ $17 chairs from 40 chairs and arrange the boys$ $Total arrangement =^{40} C_{17} \times (17 - 1)!$ $But you said : Total arrangement =^{40 - 1} P_{17 - 1}$ $I just want to understand sir .$ $(2) I still {don}^{'} t get the 2^{15} sir . 15! \times 2^{15} . please explain sir . Thanks$
	Commented by mr W last updated on 14/Sep/20

	$(2)$ $i n t h e 15 p a i r s o f s e a t s e a c h b o y c a n$ $s e l e c t o n e o f t h e t w o s e a t s, s o t o t a l l y$ $t h e r e a r e 2^{15} d i f f e r e n t w a y s .$
	Commented by mr W last updated on 14/Sep/20

	$(1)$ $y o u c a n r e a d a t e x t b o o k a b o u t c i r c u l a r$ $a r r a n g e m e n t .$ $m y w a y t o u n d e r s t a n d :$ $i n a c i r c u l a r a r r a n g e m e n t i t {d o e s n}^{'} t$ $m a t t e r w h e r e t h e a r r a n g e m e n t$ $b e g i n n s . s o w e c a n f i x a b o y i n a s e a t .$ $w e o n l y n e e d t o d e t e r m i n e i n h o w$ $m a n y w a y s w e c a n a r r a n g e t h e$ $r e m a i n i n g 16 b o y s i n t h e r e m a i n i n g$ $39 s e a t s, t h a t i s P_{16}^{39} .$
	Commented by I want to learn more last updated on 14/Sep/20

	$Thank you sir, i understand now$
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