How many ways to arrnge the letters ABCCCDEFG (1) in general . (2) all 3 Cs must be together (3) only 2 Cs must be together (4) no 2 or 3 Cs be together (5) no letter still in its original place .


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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

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