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Question Number 189049 by mr W last updated on 11/Mar/23

Commented by mr W last updated on 13/Mar/23

$t h i s i s a n o l d q u e s t i o n (Q 164560) .$ $m a y b e m e a n w h i l e s o m e b o d y h a s g o t$ $a n e w s o l u t i o n .$
Commented by nikif99 last updated on 11/Mar/23

	Answered by nikif99 last updated on 11/Mar/23

	$132$ $A t f i r s t, I t r i e d t o s o l v e i t g r a p h i c a l l y$ $[s e e a t t a c h e d; e v e r y r o w i s a q u e u e$ $o f g i r l s (p i n k) a n d b o y s (b l u e)] n o t$ $c r e a t i n g p r o b l e m t o t h e c a s h i e r .$ $T h e n, I t r i e d t o g e n e r a l i s e t h e s o l u t i o n$ $(5 g + 5 b : 42; 4 g + 4 b : 14) w i t h o u t s u c c e s .$ $L a t e r, I n o t i c e d a c o m m e n t i n a (g r e e k)$ $a r t i c l e a b o u t t h e p r i n c i p l e o f r e f l e c t i o n$ $i n c o m b i n a t o r i c s w i t h t h e s o l u t i o n :$ $T h e p r o b a b i l i t y f o r n o t p r o b l e m t o t h e$ $c a s h i e r i s 1 / (n + 1) .$ $C o n s i d e r i n g a l l p o s s i b l e a r r a n g e m e n t s$ $i n q u e u e i s C_{n}^{2 n}, t h e a c c e p t e d q u e u e s a r e$ $\frac{1}{n + 1} C_{n}^{2 n} = \frac{1}{7} \cdot 924 = 132$
	Commented by nikif99 last updated on 11/Mar/23

	Commented by nikif99 last updated on 11/Mar/23

	Commented by manxsol last updated on 12/Mar/23

	$i n t e r e s a n t e . c o m a n d i n$ $e x c e l c r e a t i o n t a b l e . p l e a s e o r$ $m a c r o . t h a n k S r n i k k i .$ $a h l i n k a r t i c l e ?$
	Commented by mr W last updated on 12/Mar/23

	$\underset{―}{m y c o n s i d e r a t i o n :}$ $a t a n y p o i n t i n t h e q u e u e t h e t o t a l$ $n u m b e r o f g i r l s m u s t b e m o r e t h a n$ $o r e q u a l t o t h e t o t a l n u m b e r o f b o y s .$ $w e c a n d i s p l a y t h i s a s a s t e p p a t h l i k e$ $t h e f o l l o w i n g o n e . i f s u c h a p a t h l i e s$ $c o m p l e t e l y a b o v e t h e d i a g o n a l$ $(s h a d e d z o n e), t h e n t h e c a s h i e r h a s$ $n o p r o b l e m w i t h c h a n g e .$
	Commented by mr W last updated on 12/Mar/23

	Commented by mr W last updated on 12/Mar/23

	$t h i s e x a m p l e s h o w s a p a t h f o r t h e$ $c a s e w i t h 6 g i r l s a n d 6 b o y s . i t$ $r e p r e s e n t s t h e q u e u e : 2 g i r l s, 1 b o y,$ $1 g i r l, 2 b o y s, 2 g i r l s, 1 b o y, 1 g i r l a n d$ $f i n a l l y 2 b o y s .$ $g e n e r a l l y w i t h n g i r l s a n d n b o y s, t h e$ $n u m b e r o f v a l i d p a t h s i s$ $C_{n} = \frac{C_{n}^{2 n}}{n + 1} = \frac{(2 n)!}{(n + 1)! n!}$ $w h i c h i s t h e C a t a l a n n u m b e r .$
	Commented by mr W last updated on 12/Mar/23

	$n i k i f s i r h a s g o t 14 f o r n = 4 a n d 42$ $f o r n = 5 a n d 132 f o r n = 6, t h e y a r e$ $a l l c o r r e c t a n d a r e t h e c o r r e s p o n d i n g$ $C a t a l a n n u m b e r s .$
	Commented by nikif99 last updated on 12/Mar/23

	$W o n d e r f u l l e s s o n . T h a n k y o u S i r!$
	Commented by manxsol last updated on 12/Mar/23

	$T h a n k y o u v e r y m u c h$ $S i r . W a n d S r . N i k k i f f o r$ $y o u r e f f o r t a n d f o r s h a r i n g$
	Commented by nikif99 last updated on 12/Mar/23

	$@ m a n x s o l$ $- n o m a c r o u s e d i n e x c e l . R o w s$ $c r e a t e d m a n u a l l y i n b l o c k s, f o l l o w i n g$ $c o m b i n a t i o n p a t t e r n .$ $- I h a v e n o t t h e a r t i c l e, j u s t a p i c t u r e$ $k e p t . T h e p r o b l e m i s p r e s e n t e d m o r e$ $c o m p l i c a t e d . T e x t s a y s :$ $T h e r e i s a q u e u e o f 2 n p e r s o n s o v e r t h e$ $c a s h i e r o f a t h e a t e r . H a l f o f t h e m h a v e$ $o n l y b a n k n o t e s o f 1000, w h i l e t h e o t h e r$ $h a l f h a v e s o m e 1000 s a n d o n l y o n e$ $b a n k n o t e o f 500 . T i c k e t s c o s t 2500 a n d$ $3500 . T h e c a s h i e r h a s n o m o n e y a t$ $t h e b e g i n n i n g . I t i s c l e a r t h a t t h e y a r e$ $(\begin{matrix} 2 n \\ n \end{matrix}) w a y s t o a r r a n g e i n q u e u e .$ $I n h o w m a n y o f t h e s e w a y s t h e c a s h i e r$ $h a s a l w a y s 500, s o n o p r o b l e m o c c u r s ?$ $F r a m e b o t t o m r i g h t : p r o b a b i l i t y f o r$ $n o p r o b l e m : 1 / (n + 1) .$
	Commented by nikif99 last updated on 12/Mar/23

	Commented by manxsol last updated on 12/Mar/23

	$t h a n s S r . n i k k i f o r d e d i c a t i n g$ $y o u r t i m e f o r m y q u e s t i o n$
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