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Question Number 95723 by i jagooll last updated on 27/May/20

Commented by bobhans last updated on 27/May/20
the case is ssma with there are 5 women 2 men sitting in 7 chairs with no 2 men sitting side by side
	Answered by Rio Michael last updated on 27/May/20

	$the 5 girls can stand in a row in 5! ways$ $let the girls be labeled as below$ $G_{1} G_{2} G_{3} G_{4} G_{5}$ $the first boy coming in has 6 vacancies$ $the second has 5 vacanciies$ $the third has 4 vacancies so$ $\Rightarrow number of ways = 5! \times 6 \times 5 \times 4 = 14 400 ways$
	Commented by mr W last updated on 27/May/20

	$14400 i s c o r r e c t! R i o M i c h a e l s i r :$ $w h e r e d o y o u t h i n k y o u a r e w r o n g ?$
	Commented by i jagooll last updated on 27/May/20

	$your answer not include G G BGBGGB ?$
	Commented by Rio Michael last updated on 28/May/20

	$well sir i was in a hurry on that problem$ $when the others dropped thier solution$ $i thought i misread it .$
	Answered by bobhans last updated on 27/May/20

	$totally arrangement = 8!$ $2 boy stand are together = C_{2}^{3} . 3! 6!$ $then = 8! - 6.6! = 6! \times 50$
	Answered by mr W last updated on 27/May/20

	$M E T H O D I$ $b e t w e e n t w o b o y s t h e r e m u s t b e a t l e a s t$ $o n e g i r l .$ $w e a r r a n g e a t f i r s t t h e 5 g i r l s, t h e r e$ $a r e 5! w a y s .$ $◻ G ◻ G ◻ G ◻ G ◻ G ◻$ $t h e 3 b o y s c a n n o w t a k e t h r e e o f$ $6 p o s i t i o n s (◻) . t h e r e a r e C_{3}^{6} \times 3! w a y s .$ $s o t o t a l l y t h e r e a r e C_{3}^{6} \times 3! \times 5! = 14400$ $w a y s .$ $M E T H O D I I$ $t o a r r a n g e 8 c h i l d r e n t h e r e a r e 8!$ $w a y s .$ $t o a r r a n g e t h e m s u c h t h a t t h r e e b o y s$ $a r e t o g e t h e r, t h e r e a r e 6! \times 3! w a y s .$ $t o a r r a n g e t h e m s u c h t h a t t w o a n d$ $o n l y t w o b o y s a r e n e x t t o e a c h o t h e r,$ $t h e r e a r e C_{2}^{3} \times (7! - 2 \times 6!) \times 2! w a y s .$ $t o t a l v a l i d a r r a n g e m e n t s :$ $8! - C_{2}^{3} \times (7! - 2 \times 6!) \times 2! - 6! \times 3! = 14400$
	Commented by john santu last updated on 28/May/20

	$oo yes . . . your right . i^{'} m using method II$ $but forgot something . ok thanks you$
	Commented by bobhans last updated on 28/May/20

	$how with ◼ GG ◼ G ◼ GG$ $with ◼ GGG ◼ G ◼ G ?$ $your answer included it ?$
	Commented by mr W last updated on 28/May/20

	$y e s, i n c l u d e d .$ $t h e b o y s c a n t a k e a n y t h r e e p o s i t i o n s$ $f r o m t h e 6 ◻ i n :$ $◻ G ◻ G ◻ G ◻ G ◻ G ◻$ $e x a m p l e s :$ $◼ G ◻ G ◼ G ◼ G ◻ G ◻$ $◼ G ◻ G ◻ G ◼ G ◼ G ◻$
	Commented by i jagooll last updated on 28/May/20

	$thank you sir$
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