sequence of string said to be orderly if element index i different to i+1 for example aba has orderly value 2 abab has orderly value 3 abaabb has orderly value 3 if there are 7 a and 13 b example aaaaaaabbbbbbbbbbbbb has orderly value 1 what is the mean of its orderly value for all possible sequences?


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Question Number 195666 by uchihayahia last updated on 07/Aug/23

$s e q u e n c e o f s t r i n g s a i d t o b e o r d e r l y$ $i f e l e m e n t i n d e x i d i f f e r e n t t o i + 1$ $f o r e x a m p l e$ $a b a h a s o r d e r l y v a l u e 2$ $a b a b h a s o r d e r l y v a l u e 3$ $a b a a b b h a s o r d e r l y v a l u e 3$ $i f t h e r e a r e 7 a a n d 13 b$ $e x a m p l e$ $a a a a a a a b b b b b b b b b b b b b h a s o r d e r l y v a l u e 1$ $w h a t i s t h e m e a n o f i t s o r d e r l y v a l u e$ $f o r a l l p o s s i b l e s e q u e n c e s ?$
Commented by mr W last updated on 09/Aug/23

$i g o t 9.1 i n a t o u g h w a y, s e e b e l o w .$ $d o y o u a l s o h a v e a s o l u t i o n ?$
Commented by uchihayahia last updated on 09/Aug/23

$i {d o n}^{'} t, i t r i e d s o l v i n g i t u s i n g p y t h o n$ $t o o m u c h t i m e n e e d e d$
	Answered by mr W last updated on 09/Aug/23

	$7 ‘ ‘ a^{″} a n d 13 ‘ ‘ b^{″}$ $n = o r d e r l y v a l u e$ $n_{m i n} = 1$ $n_{m a x} = 14$ $i n f o l l o w i n g$ $A m e a n s a b o x c o n t a i n i n g o n e o r$ $m o r e l e t t e r s ‘ ‘ a^{″}$ $B m e a n s a b o x c o n t a i n i n g o n e o r$ $m o r e l e t t e r s ‘ ‘ b^{″}$ $n = 1 :$ $A ∣ B \Rightarrow 1 w a y$ $B ∣ A \Rightarrow 1 w a y$ $- - - 2$ $n = 2 :$ $A ∣ B ∣ A \Rightarrow 6 \times 1 = 6 w a y s$ $B ∣ A ∣ B \Rightarrow 1 \times 12 = 12 w a y s$ $- - - 18$ $n = 3 :$ $A ∣ B ∣ A ∣ B \Rightarrow 6 \times 12 = 72 w a y s$ $B ∣ A ∣ B ∣ A \Rightarrow 6 \times 12 = 72 w a y s$ $- - - 144$ $n = 4 :$ $A ∣ B ∣ A ∣ B ∣ A \Rightarrow 15 \times 12 = 180 w a y s$ $B ∣ A ∣ B ∣ A ∣ B \Rightarrow 6 \times 66 = 396 w a y s$ $- - - 576$ $n = 5 :$ $A ∣ B ∣ A ∣ B ∣ A ∣ B \Rightarrow 15 \times 66 = 990 w a y s$ $B ∣ A ∣ B ∣ A ∣ B ∣ A \Rightarrow 15 \times 66 = 990 w a y s$ $- - - 1980$ $n = 6 :$ $A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A \Rightarrow 20 \times 66 = 1320 w a y s$ $B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B \Rightarrow 15 \times 220 = 3300 w a y s$ $- - - 4620$ $n = 7 :$ $A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B \Rightarrow 20 \times 220 = 4400 w a y s$ $B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A \Rightarrow 20 \times 220 = 4400 w a y s$ $- - - 8800$ $n = 8 :$ $A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A \Rightarrow 15 \times 220 = 3300 w a y s$ $B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B \Rightarrow 20 \times 495 = 9900 w a y s$ $- - - 13200$ $n = 9 :$ $A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B \Rightarrow 15 \times 495 = 7425 w a y s$ $B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A \Rightarrow 15 \times 495 = 7425 w a y s$ $- - - 14850$ $n = 10 :$ $A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A \Rightarrow 6 \times 495 = 2970 w a y s$ $B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B \Rightarrow 15 \times 792 = 11880 w a y s$ $- - - 14850$ $n = 11 :$ $A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B \Rightarrow 6 \times 792 = 4752 w a y s$ $B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A \Rightarrow 6 \times 792 = 4752 w a y s$ $- - - 9504$ $n = 12 :$ $A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A \Rightarrow 1 \times 792 = 792 w a y s$ $B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B \Rightarrow 6 \times 924 = 5544 w a y s$ $- - - 6336$ $n = 13 :$ $A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B \Rightarrow 1 \times 924 = 924 w a y s$ $B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A \Rightarrow 1 \times 924 = 924 w a y s$ $- - - 1848$ $n = 14 :$ $B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B ∣ A ∣ B \Rightarrow 1 \times 792 = 792 w a y s$ $- - - 792$ $m e a n = \frac{2 \times 1 + 18 \times 2 + 144 \times 3 + 576 \times 4 + 1980 \times 5 + 4620 \times 6 + 8800 \times 7 + 13200 \times 8 + 14850 \times 9 + 14850 \times 10 + 9504 \times 11 + 6336 \times 12 + 1848 \times 13 + 792 \times 14}{2 + 18 + 144 + 576 + 1980 + 4620 + 8800 + 13200 + 14850 + 14850 + 9504 + 6336 + 1848 + 792}$ $= \frac{705 432}{77 520} = 9.1 ✓$ $c h e c k :$ $n u m b e r o f t o t a l p o s s i b i l i t i e s$ $= \frac{20!}{7! \times 13!} = 77 520 ✓$
	Commented by uchihayahia last updated on 09/Aug/23

	$t h a n k s y o u, i^{'} l l d o m y b e s t t o u n d e r s t a n d y o u r$ $a n s w e r . s t i l l s t u d y i n g b a s i c c o m b i n a t r i c s$
	Commented by mr W last updated on 09/Aug/23

	$i^{'} l l g i v e s o m e e x p l a n a t i o n f o r m y$ $s o l u t i o n .$ $e x a m p l e : t h e o r d e r l y v a l u e i s n = 4 .$ $t h a t m e a n s t h e r e a r e 4$ $p l a c e s w h e r e a ‘ ‘ a^{″} a n d a ‘ ‘ b^{″} a r e$ $n e x t t o e a c h o t h e r . s u c h a p l a c e i s$ $m a r k e d a s ‘ ‘ ∣^{″} . w e h a v e t w o c a s e s :$ $c a s e 1 : A ∣ B ∣ A ∣ B ∣ A$ $t o d i s t r i b u t e 7 ‘ ‘ a^{″} i n 3 b o x e s t h e r e a r e$ $C_{7 - 1}^{3 - 1} = 15 w a y s$ $t o d i s t r i b u t e 13 ‘ ‘ b^{″} i n 2 b o x e s t h e r e a r e$ $C_{13 - 1}^{2 - 1} = 12 w a y s$ $\Rightarrow t o t a l l y 15 \times 12 = 180 w a y s$ $c a s e 2 : B ∣ A ∣ B ∣ A ∣ B$ $t o d i s t r i b u t e 7 ‘ ‘ a^{″} i n 2 b o x e s t h e r e a r e$ $C_{7 - 1}^{2 - 1} = 6 w a y s$ $t o d i s t r i b u t e 13 ‘ ‘ b^{″} i n 3 b o x e s t h e r e a r e$ $C_{13 - 1}^{3 - 1} = 66 w a y s$ $\Rightarrow t o t a l l y 6 \times 66 = 396 w a y s$ $t h e r e f o r e t h e r e a r e 180 + 396 = 576$ $p o s s i b i l i t i e s f o r t h e o r d e r l y v a l u e 4 .$
	Commented by uchihayahia last updated on 11/Aug/23

	$t h a n k s, s t i l l l o t o f w o r k i g u e s s . i a s k e d$ $m y f r i e n d a n d t o l d m e 9.1 i s t h e c o r r e c t$ $a n s w e r$
	Commented by mr W last updated on 11/Aug/23

	$a n y w a y m y a n s w e r 9.1 i s c o r r e c t .$ $c a n y o u r f r i e n d t e l l u s h o w h e s o l v e d ?$
	Commented by uchihayahia last updated on 12/Aug/23

	$h e {d i d n}^{'} t t e l l m e, h e s a i d t h e a n s w e r$ $i s l o n g a n d l a b o r o u s . b u t t h e i d e a i s t h e s a m e$
	Commented by mr W last updated on 12/Aug/23

	$i a l s o t h i n k t h e r e i s n o m o r e s i m p l e$ $w a y t h a n t h a t o f m i n e .$
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