the family A has 5 members and the family B has 4 members. there are 6 personsfrom other families. in how many ways can you arrange these 15 persons around a round table such that no member from family A and no member from family B are next to each other?


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Question Number 195964 by mr W last updated on 15/Aug/23

$t h e f a m i l y A h a s 5 m e m b e r s a n d t h e$ $f a m i l y B h a s 4 m e m b e r s . t h e r e a r e$ $6 p e r s o n s f r o m o t h e r f a m i l i e s .$ $i n h o w m a n y w a y s c a n y o u a r r a n g e$ $t h e s e 15 p e r s o n s a r o u n d a r o u n d t a b l e$ $s u c h t h a t n o m e m b e r f r o m f a m i l y A$ $a n d n o m e m b e r f r o m f a m i l y B a r e$ $n e x t t o e a c h o t h e r ?$
	Answered by mr W last updated on 15/Aug/23

	Commented by mr W last updated on 15/Aug/23

	$w e a r r a n g e a t f i r s t t h e 6 p e o p l e f r o m$ $o t h e r f a m i l i e s (g r e e n s e a t s) o n$ $t h e t a b l e . t h e r e a r e 5! w a y s t o d o t h i s .$ $s a y ♠ s t a n d s f o r a s e a t f o r a m e m b e r$ $f r o m f a m i l y A a n d ♣ s t a n d s f o r a$ $s e a t f o r a m e m b e r f r o m f a m i l y B .$ $e a c h o f t h e 6 p l a c e s (b l u e b o x e s)$ $b e t w e e n t h e g r e e n s e a t s c a n b e f i l l e d$ $w i t h :$ $n o t h i n g (e m p t y) \Leftrightarrow w e a d d 0 t o i t$ $o r a s e a t ♠ \Leftrightarrow w e a d d 1 t o i t$ $o r 2 s e a t s ♠ ♠ \Leftrightarrow w e a d d 2 t o i t$ $o r 3 s e a t s ♠ ♠ ♠ \Leftrightarrow w e a d d 3 t o i t$ $o r 4 s e a t s ♠ ♠ ♠ ♠ \Leftrightarrow w e a d d 4 t o i t$ $o r 5 s e a t s ♠ ♠ ♠ ♠ ♠ \Leftrightarrow w e a d d 5 t o i t$ $o r a s e a t ♣ \Leftrightarrow w e a d d 100 t o i t$ $o r 2 s e a t s ♣ ♣ \Leftrightarrow w e a d d 200 t o i t$ $o r 3 s e a t s ♣ ♣ ♣ \Leftrightarrow w e a d d 300 t o i t$ $o r 4 s e a t s ♣ ♣ ♣ ♣ \Leftrightarrow w e a d d 400 t o i t$ $i n a l l 6 b o x e s w e s h o u l d t o t a l l y p u t$ $5 s e a t s ♠ ♠ ♠ ♠ ♠ a n d 4 s e a t s ♣ ♣ ♣ ♣ .$ $t h a t m e a n s t h e s u m o f a l l 6 b o x e s$ $s h o u l d b e 5 + 400 = 405 . s o t h e n u m b e r$ $o f w a y s t o d o t h i s i s t h e c o e f f i c i e n t$ $o f t e r m x^{405} i n t h e e x p a n s i o n o f t h e$ $g e n e r a t i n g f u n c t i o n$ ${(1 + x + x^{2} + x^{3} + x^{4} + x^{5} + x^{100} + x^{200} + x^{300} + x^{400})}^{6},$ $w h i c h i s 4626, s e e b e l o w . s i n c e t h e$ $m e m b e r s b o t h i n f a m i l y A a n d$ $f a m i l y B a r e p e r m u t a b l e, t h e n u m b e r$ $o f w a y s t o a r r a n g e t h e m e m b e r s$ $f r o m f a m i l y A a n d B i s 4626 \times 4! \times 5! .$ $t h e t o t a l y n u m b e r o f w a y s t o a r r a n g e$ $a l l p e o p l e a r o u n d t h e t a b l e i s t h u s$ $4626 \times 4! \times 5! \times 5! = 1 598 745 600$
	Commented by mr W last updated on 15/Aug/23

	Commented by MM42 last updated on 15/Aug/23

	$s i r W$ $h o w t o d r a w g e o m e t r i c s h a p e s$ $o n t h i s p a g e ? a s i n t h e f i g u r e a b o v e$
	Commented by mr W last updated on 15/Aug/23

	$t r y L E K H d i a g r a m .$
	Commented by MM42 last updated on 15/Aug/23

	$t h a n k y o u s o$
	Commented by Tawa11 last updated on 16/Aug/23

	$Great work sir .$
	Commented by York12 last updated on 16/Aug/23

	$s i r w h e r e t o l e a r n c o m b i n a t o r i c s$
	Commented by York12 last updated on 16/Aug/23

	$o r a s e a t ♠ \Leftrightarrow w e a d d 1 t o i t$ $o r 2 s e a t s ♠ ♠ \Leftrightarrow w e a d d 2 t o i t$ $o r 3 s e a t s ♠ ♠ ♠ \Leftrightarrow w e a d d 3 t o i t$ $o r 4 s e a t s ♠ ♠ ♠ ♠ \Leftrightarrow w e a d d 4 t o i t$ $o r 5 s e a t s ♠ ♠ ♠ ♠ ♠ \Leftrightarrow w e a d d 5 t o i t$ $o r a s e a t ♣ \Leftrightarrow w e a d d 100 t o i t$ $o r 2 s e a t s ♣ ♣ \Leftrightarrow w e a d d 200 t o i t$ $o r 3 s e a t s ♣ ♣ ♣ \Leftrightarrow w e a d d 300 t o i t$ $o r 4 s e a t s ♣ ♣ ♣ ♣ \Leftrightarrow w e a d d 400 t o i t$ $i n a l l 6 b o x e s w e s h o u l d t o t a l l y p u t$ $5 s e a t s ♠ ♠ ♠ ♠ ♠ a n d 4 s e a t s ♣ ♣ ♣ ♣ .$ $t h a t m e a n s t h e s u m o f a l l 6 b o x e s$ $s h o u l d b e 5 + 400 = 405$
	Commented by York12 last updated on 16/Aug/23

	$I d o n o t u n d e r s t a n d w h a t d o y o u$ $m e a n b y t h i s p a r t$
	Commented by mr W last updated on 16/Aug/23

	$y o u h a v e 5 s e a t s ♠ f o r t h e f a m i l y A .$ $i n e a c h b o x y o u c a n p u t 1 o r 2 o r 3 o r$ $4 o r 5 s e a t s ♠ . t o t a l l y y o u m u s t p u t$ $a l l 5 s e a t s ♠ . o r y o u p u t i n t h i s$ $o n l y s e a t s f o r f a m i l y B, n a m l y$ $1 o r 2 o r 3 o r 4 s e a t s ♣ . t o t a l l y y o u$ $m u s t p u t a l l 4 s e a t s ♣ .$ $i f e m p t y m e a n s 0, a ♠ m e a n s 1,$ $2 ♠ ♠ m e a n s 2, 3 ♠ ♠ ♠ m e a n s 3,$ $4 ♠ ♠ ♠ ♠ m e a n s 4, 5 ♠ ♠ ♠ ♠ ♠ m e a n s$ $5, a n d a ♣ m e a n s 100, 2 ♣ ♣ m e a n s$ $200, 3 ♣ ♣ ♣ m e a n s 300, 4 ♣ ♣ ♣ ♣$ $m e a n s 400, t h e s u m o f n u m b e r s i n$ $a l l 6 b o x e s m u s t b e 405 .$
	Commented by York12 last updated on 16/Aug/23

	$t h a n k s$
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