say you have 3 (different) books about mathematics, 4 (different) books about physics and 5 (different) books about chemistry. in how many ways can you arrange them in a shelf such that no two books from the same subject are adjacent?


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Question Number 196143 by mr W last updated on 18/Aug/23

$s a y y o u h a v e 3 (d i f f e r e n t) b o o k s$ $a b o u t m a t h e m a t i c s, 4 (d i f f e r e n t)$ $b o o k s a b o u t p h y s i c s a n d 5 (d i f f e r e n t)$ $b o o k s a b o u t c h e m i s t r y . i n h o w m a n y$ $w a y s c a n y o u a r r a n g e t h e m i n a s h e l f$ $s u c h t h a t n o t w o b o o k s f r o m t h e s a m e$ $s u b j e c t a r e a d j a c e n t ?$
	Answered by liuxinnan last updated on 19/Aug/23

	Commented by mr W last updated on 19/Aug/23

	$t h a n k s f o r t r y i n g s i r! {i t}^{'} s n o t e a s y t o$ $f o l l o w y o u r s o l u t i o n . c a n y o u p l e a s e$ $r e c h e c k y o u r c a l c u l a t i o n ? i g o t a$ $s l i g h t l y d i f f e r e n t r e s u l t .$
	Commented by mr W last updated on 19/Aug/23

	$i h a v e f o u n d y o u r m i s t a k e s :$
	Commented by mr W last updated on 19/Aug/23

	Commented by mr W last updated on 19/Aug/23

	$e r r o r 1 : M P c a n a l s o b e P M .$ $e r r o r 2 : P M P M c a n a l s o b e M P M P .$
	Commented by liuxinnan last updated on 19/Aug/23

	$I c a n n o t f i n d o u t m e e r r o t t o o$
	Commented by liuxinnan last updated on 20/Aug/23

	$t h a n k s s i r$
	Answered by mr W last updated on 19/Aug/23

	$i n f o l l o w i n g$ $♣ = a b o o k f o r p h y s i c s$ $⧫ = a b o o k f o r m a t h e m a t i c s$ $★ = a b o o k f o r c h e m i s t r y$ $w e a r r a n g e a t f i r s t t h e 5 b o o k s f o r$ $c h e m i s t r y :$ $◻ ★ ⊡ ★ ⊡ ★ ⊡ ★ ⊡ ★ ◻$ $i n e a c h o f t h e ◻ b o x e s w e c a n p u t$ $n o t h i n g, i . e . e m p t y \Leftrightarrow 0$ $♣ \Leftrightarrow 1$ $⧫ \Leftrightarrow 100$ $♣ ⧫ \Leftrightarrow 101 (2 \times)$ $♣ ⧫ ♣ \Leftrightarrow 102$ $⧫ ♣ ⧫ \Leftrightarrow 201$ $♣ ⧫ ♣ ⧫ \Leftrightarrow 202 (2 \times)$ $♣ ⧫ ♣ ⧫ ♣ \Leftrightarrow 203$ $⧫ ♣ ⧫ ♣ ⧫ \Leftrightarrow 302$ $♣ ⧫ ♣ ⧫ ♣ ⧫ \Leftrightarrow 303 (2 \times)$ $♣ ⧫ ♣ ⧫ ♣ ⧫ ♣ \Leftrightarrow 304$ $i n e a c h o f t h e ⊡ b o x e s w e c a n d o a s$ $a b o v e, b u t e m p t y i s n o t a l l o w e d .$ $s o w e g e t a g e n e r a t i n g f u n c t i o n$ ${(1 + x + x^{100} + 2 x^{101} + x^{102} + x^{201} + 2 x^{202} + x^{203} + x^{302} + 2 x^{303} + x^{304})}^{2} {(x + x^{100} + 2 x^{101} + x^{102} + x^{201} + 2 x^{202} + x^{203} + x^{302} + 2 x^{303} + x^{304})}^{4}$ $w h o s e c o e f f i c i e n t o f t e r m x^{304} i s 588 .$ $\Rightarrow t o t a l n u m b e r o f w a y s i s$ $588 \times 5! \times 4! \times 3! = 10 160 640$
	Commented by mr W last updated on 19/Aug/23

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