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-f

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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