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Question Number 135217 by benjo_mathlover last updated on 11/Mar/21

Commented by mr W last updated on 11/Mar/21

$p l e a s e e x p l a i n w h a t y o u m e a n w i t h$ $‘ ‘ t h e r e i s e x a c t l y o n l y a c i r c l e t h a t$ $h a s 6 {g i r l s}^{″}!$ $w h e n a c i r c l e h a s e x a c t l y 6 g i r l s,$ $t h e n t h e o t h e r c i r c l e a l s o h a s e x a c t l y$ $6 g i r l s .$
Commented by benjo_mathlover last updated on 11/Mar/21

one of the circles exactly 6 girls means there shouldn't be a boy among them
Commented by mr W last updated on 11/Mar/21

$t h a t m e a n s o n e c i r c l e h a s 6 g i r l s,$ $a n d t h e o t h e r c i r c l e h a s 6 g i r l s a n d$ $8 b o y s .$
Commented by benjo_mathlover last updated on 11/Mar/21

$y e s s i r$
Commented by mr W last updated on 11/Mar/21

$t h e q u e s t i o n i s n o t e a s y . d o y o u h a v e$ $t h e a n s w e r ?$
	Answered by mr W last updated on 12/Mar/21

	$s t e p 1 :$ $i n h o w m a n y w a y s c a n 8 b o y s a n d$ $12 g i r l s b e d i v i d e d i n t o t w o g r o u p s$ $s u c h t h a t e a c h g r o u p h a s a t l e a s t 6$ $p e o p l e ?$ $s a y i n o n e g r o u p t h e n u m b e r o f b o y s$ $i s b a n d t h e n u m b e r o f g i r l s i s g,$ $6 ⩽ b + g ⩽ 14$ $t o s e l e c t b f r o m 8 b o y s t h e r e a r e C_{b}^{8} w a y s$ $t o s e l e c t g f r o m 12 g i r l s t h e r e a r e C_{g}^{12} w a y s$ $b : (1 + C_{1}^{8} x + C_{2}^{8} x^{2} + C_{3}^{8} x^{3} + . . . + C_{8}^{8} x^{8})$ $g : (1 + C_{1}^{12} x + C_{2}^{12} x^{2} + C_{3}^{12} x^{3} + . . . + C_{12}^{12} x^{12})$ $t h e s u m o f c o e f . o f t e r m s x^{6} t o x^{14} i n$ $(1 + C_{1}^{8} x + C_{2}^{8} x^{2} + C_{3}^{8} x^{3} + . . . + C_{8}^{8} x^{8}) (1 + C_{1}^{12} x + C_{2}^{12} x^{2} + C_{3}^{12} x^{3} + . . . + C_{12}^{12} x^{12})$ $i s t h e r e s u l t .$ $x^{6} : 38 760$ $x^{7} : 77 520$ $x^{8} : 125 970$ $x^{9} : 167 960$ $x^{10 :} 184 756$ $x^{11} : 167 960$ $x^{12} : 125 970$ $x^{13} : 77 520$ $x^{14} : 38 760$ $Σ : 1 005 176$
	Commented by mr W last updated on 12/Mar/21

	$s t e p 2 :$ $a s s t e p 1, b u t e a c h g r o u p m u s t h a v e$ $b o t h b o y a n d g i r l, t h a t m e a n s o n e$ $g r o u p m u s t h a v e a t l e a s t o n e b u t$ $a t m o s t 7 b o y s, a n d a t l e a s t o n e b u t$ $a t m o s t 11 g i r l s .$ $t h e s u m o f c o e f . o f t e r m s x^{6} t o x^{14} i n$ $(C_{1}^{8} x + C_{2}^{8} x^{2} + C_{3}^{8} x^{3} + . . . + C_{7}^{8} x^{7}) \times (C_{1}^{12} x + C_{2}^{12} x^{2} + C_{3}^{12} x^{3} + . . . + C_{11}^{12} x^{11})$ $i s t h e r e s u l t .$ $x^{6} : 37 808$ $x^{7} : 76 720$ $x^{8} : 125 474$ $x^{9} : 167 728$ $x^{10} : 184 624$ $x^{11} : 167 728$ $x^{12} : 125 474$ $x^{13} : 76 720$ $x^{14} : 37 808$ $Σ : 1 000 084$
	Commented by mr W last updated on 11/Mar/21

	$s t e p 3 :$ $o n e g r o u p h a s o n l y 6 g i r l s .$ $r e s u l t i s C_{6}^{12} = 924$
	Commented by mr W last updated on 11/Mar/21

	$t h e r e q u e s t e d p r o b a b i l i t y i s$ $p = \frac{1 000 084 + 924}{1 005 176} = \frac{125 126}{125 647} \approx 0.995 853$
	Commented by mr W last updated on 11/Mar/21

	Commented by mr W last updated on 11/Mar/21

	Commented by benjo_mathlover last updated on 13/Mar/21

	$w a w = . ====$
	Commented by mr W last updated on 13/Mar/21

	$y o u c o u l d c o m f i r m t h e a n s w e r s i r ?$
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