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Question Number 218076 by malwan last updated on 28/Mar/25

H o w m a n y w a y s t o a r r n g e

t h e l e t t e r s A B C C C D E F G

(1) i n g e n e r a l .

(2) a l l 3 C s m u s t b e t o g e t h e r

(3) o n l y 2 C s m u s t b e t o g e t h e r

(4) n o 2 o r 3 C s b e t o g e t h e r

(5) n o l e t t e r s t i l l i n i t s

o r i g i n a l p l a c e .

Answered by efronzo1 last updated on 29/Mar/25

(1) \frac{9!}{3!}

(2) 7!

Answered by mr W last updated on 29/Mar/25

(1)

\frac{9!}{3!} = 60480

(2)

7! = 5040

o r 6! \times 7 = 5040

(3)

A B D E F G

6! \times 7 \times 6 = 30240

t h e t w o C s a n d a C m u s t b e s e p a r a t e d

b y o t h e r l e t t e r s . t h e y c a n b e p l a c e d

i n t o t h e 7 r e d b o x e s .

t o a r r a n g e t h e 6 o t h e r l e t t e r s, t h e r e

a r e 6! w a y s, t o p l a c e t w o C s a n d a C,

t h e r e a r e 7 \times 6 w a y s .

t o t a l l y 6! \times 7 \times 6

(4)

6! \times 7 \times 6 \times 5 / 3! = 25200

s e e a l s o a b o v e . t h e t h r e e C s m u s t

b e s e p a r a t e d b y t h e o t h e r l e t t e r s .

t o p l a c e t h e t h r e e C s i n t o t h e 7

b o x e s, t h e r e a r e C_{7}^{3} = 7 \times 6 \times 5 / 3!

w a y s .

t o t a l l y 6! \times 7 \times 6 \times 5 / 3!

(5)

A B C C C D E F G

E F A B D G C C C

C_{6}^{3} \times 3! \times (\frac{6!}{3!} - 1 - 3 \times 3 - 3 \times 13) = 8520

u n d e r t h e t h r e e C s o t h e r l e t t e r s

m u s t b e p l a c e d . t o s e l e c t 3 l e t t e r s

a n d p l a c e t h e m u n d e r t h e C s t h e r e

a r e C_{6}^{3} \times 3! = 120 w a y s . s a y

A, B, D a r e s e l e c t e d . t o a r r a n g e t h e

r e m a i n i n g 6 l e t t e r s (e g . E F G C C C)

t h e r e a r e t o t a l l y \frac{6!}{3!} w a y s . b u t t h e r e

a r e a r r a n g e m e n t s i n w h i c h E F G

a r e p l a c e d u n d e r t h e m s e l v e s .

a l l t h r e e l e t t e r s (E F G) a r e i n f a l s e

p o s i t i o n s : t h e r e i s o n e p o s s i b i l i t y .

t w o l e t t e r s (e g . E F) a r e i n f a l s e

p o s i t i o n s : t h e r e a r e 3 \times 4 p o s s i b i l i t i e s .

b u t t h i s a l s o i n c l u d e s 3 t i m e s t h e

c a s e f r o m a b o v e . t h a t m e a n s t h e r e

a r e 3 \times 3 p o s s i b i l i t i e s f o r o n l y t w o

l e t t e r s i n f a l s e p o s i t i o n s .

o n e l e t t e r (e g . E) i n f a l s e p o s i t i o n :

t h e r e a r e 3 \times 5 \times 4 p o s s i b i l i t i e s . b u t

t h i s a l s o i n c l u d e s 7 t i m e s t h e

c a s e s a b o v e . t h a t m e a n s t h e r e a r e

3 \times 13 p o s s i b i l i t i e s f o r o n l y o n e

l e t t e r i n f a l s e p o s i t i o n .

t o t a l l y C_{6}^{3} \times 3! \times (\frac{6!}{3!} - 1 - 3 \times 3 - 3 \times 13)

Commented by malwan last updated on 29/Mar/25

t h a n k s a l o t, m r W

Commented by mr W last updated on 29/Mar/25

i^{'} v e a d d e d s o m e e x p l a n a t i o n .

Commented by malwan last updated on 29/Mar/25

y o u r a n s w e r m a y b e l i k e t h i s

\frac{6!}{3!} \times (\frac{6!}{3!} - 1 - 3 \times 3 - 3 \times 17 + 12)

= 8520

b u t, w h y + 12 ?

I h o p e t h a t y o u w i l l f i n d i t

t h a n k s a g a i n ⋚

Commented by mr W last updated on 29/Mar/25

a n s w e r f o r (5) c a n a l s o b e d e t e r m i n e d

a p p l y i n g “ r o o k {p o l y n o m i a l}^{″}

\frac{51120}{3!} = 8520

Commented by mr W last updated on 29/Mar/25

Commented by mr W last updated on 29/Mar/25

i^{'} v e g o t t h e r i g h t a n s w e r .

i t i s - 3 \times 13, n o t - 3 \times 17 .

Commented by malwan last updated on 29/Mar/25

t h a n k y o u s o m u c h s i r

\underset{―}{\underset{⏟}{}}

Commented by mr W last updated on 30/Mar/25

t h i s i s t h e e x p l a n a t i o n f o r w h y

- 3 \times 13 :

s a y w e h a v e p l a c e d A B D u n d e r C C C

A B C C C D E F G

A B D

n o w w e t r y t o p l a c e E F G C C C i n t o

t h e 6 b o x e s . t o t a l l y t h e r e a r e \frac{6!}{3!}

w a y s . b u t t h e y i n c l u d e a l s o f o l l o w i n g

c a s e s :

1) a l l E F G i n f a l s e p o s i t i o n s

C C A B D C \underset{―}{E F G}

\Rightarrow j u s t o n e t i m e

2) o n l y t w o o f E F G i n f a l s e p o s i t i o n s

i . e . E G o r E G o r F G i n f a l s e p o s i t i o n s

s a y E G i n f a l s e p o s i t i o n s :

A B D \underset{―}{E F}

G c a n b e i n w h i t e b o x e s, 3 p o s s i b i l i t i e s .

t o t a l l y 3 \times 3 p o s s i b i l i t i e s .

3) o n l y o n e o f E F G i n f a l s e p o s i t i o n

i . e . E o r F o r G i n f a l s e p o s i t i o n

s a y E i n f a l s e p o s i t i o n :

A B D \underset{―}{E}

t h e r e a r e 5 \times 4 = 20 p o s s i b i l i t i e s

t o p l a c e F G a n d C s . b u t t h i s i n c l u d e s

A B D \underset{―}{E F G} \Rightarrow 1 t i m e

A B D \underset{―}{E F} \Rightarrow 3 t i m e s (G i n)

A B D \underset{―}{E} \underset{―}{G} \Rightarrow 3 t i m e s (F i n)

t h e r e f o r e w e h a v e 20 - 7 = 13 c a s e s

w i t h o n l y E i n f a l s e p o s i t i o n .

Commented by malwan last updated on 31/Mar/25

T h a n k y o u t r i l l i o n ⋚

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