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Question Number 43031 by aseerimad last updated on 06/Sep/18

Commented by MrW3 last updated on 06/Sep/18

$8! \times (9 + C_{2}^{9}) = 8! \times 45 = 5 \times 9! = 1814400$
Commented by MrW3 last updated on 07/Sep/18

$E x p l a n a t i o n :$ $W e a r r a n g e a t f i r s t t h e o t h e r 8 s p e a k e r s,$ $t h e r e a r e 8! w a y s t o d o t h i s .$ $T h e r e a r e t w o m e t h o d s t o a r r a n g e t h e$ $s p e a k e r s A a n d B t h e n :$ $m e t h o d 1 : A a n d B a r e n e x t t o e a c h o t h e r . T h e r e$ $a r e 9 p o s s i b i l i t i e s . E . g . X X X X X X B A X X,$ $o r X X X X X X X X B A o r B A X X X X X X X X .$ $m e t h o d 2 : A a n d B a r e s e p a r a t e d b y a t$ $l e a s t o n e o f t h e o t h e r 8 p e o p l e . W e$ $c a n p l a c e A a n d B i n 9 p o s s i b l e p o s i t i o n s .$ $T h e r e a r e C_{2}^{9} w a y s . E . g . X X B X X X X A X X,$ $o r X X X X X X X B X A o r B X X X X X X X X A .$ $T h u s t h e r e s u l t i s$ $8! \times (9 + C_{2}^{9}) = 8! \times 45 = 5 \times 9!$ $G e n e r a l l y i n c a s e o f n p e o p l e :$ $(n - 2)! \times [(n - 1) + C_{2}^{n - 1}]$ $= (n - 2)! \times [(n - 1) + \frac{(n - 1) (n - 2)}{2}]$ $= \frac{n (n - 1) (n - 2)!}{2}$ $= \frac{n!}{2}$
	Answered by tanmay.chaudhury50@gmail.com last updated on 06/Sep/18

	$a r r a n g e m e n t o f s p e a k e r l i k e$ $s u p o s e 1) B i s f i r s t s p e a k e r t h e n A c a n b e a n y$ $p o s i t i o n o u t o f r e m a i n i n g 9 s o$ $p e r m u t a t i o n i s 1 \times 9 \times 8!$ $h e r e 1 f o r B a n d 8! r e m a i n i n g 8 s p e a k e r$ $2) w h e n B s e c o n d s p e a k e r A c a n b e 3 r d 4 t h . . . 10 t h$ $s o 1 \times 8 \times 8!$ $3) w h e n B t h i r d s p e a k e r A c a n b e 4 t h 5 t h . . . 10 t h$ $1 \times 7 \times 8!$ $4) w h e n B 4 t h s p e a k e r A c a n b e 5 t h 6 t h . . . 10 t h$ $1 \times 6 \times 8!$ $5) w h e n B 5 t h s o e a k e r A c a n b e 6 t h 7 t h . . . 10 t h = 1$ $1 \times 5 \times 8!$ $6) w h e n B 6 t h s p e a k e r A c a n b e 7 t h 8 t h . . 10 t h$ $1 \times 4 \times 8!$ $7) w h e n B 7 t h s p e a k e r A c a n b e 8 t h 9 t h 10 t h$ $1 \times 3 \times 8!$ $8) w h e n B 8 t h s p e a k e r A [c a n b e [9 t h 10 t h$ $1 \times 2 \times 8!$ $9) w h e n B 9 t h s p e a k e r A c a n b e 10 t h$ $1 \times 1 \times 8!$ $s o a n s i s$ $1 \times 8! (9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1)$ $= 1 \times 8! \times \frac{9}{2} {2 \times 9 + (9 - 1) \times - 1}$ $= 1 \times 8! \times \frac{9}{2} (18 - 8)$ $= 8! \times 9 \times 5$ $= 9! \times 5$
	Answered by MrW3 last updated on 06/Sep/18

	$H e r e i s a n e a s y s o l u t i o n :$ $T o a r r a n g e 10 p e o p l e t h e r e a r e 10! w a y s .$ $T h e n u m b e r o f a r r a n g e m e n t s w i t h$ $A b e f o r e B i s t h e s a m e a s t h e n u m b e r$ $o f a r r a n g e m e n t s w i t h A b e h i n d B .$ $T h u s t h e s o l u t i o n i s \frac{10!}{2} = 1814400 .$
	Commented by ajfour last updated on 07/Sep/18

	$e x c e l l e n t S i r .$
	Commented by tanmay.chaudhury50@gmail.com last updated on 07/Sep/18

	$y e s s i r . . . t h a n k y o u . . .$
	Commented by malwaan last updated on 07/Sep/18

	$thank you sir$
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