Number of ways..n differrnt things be distributed in r identical boxes so as 1)empty box is allowed 2)empty box not allowed Number of ways...n identical things be distributed to r identical boxes so as 1)empty box allowed 2)empty box not allowed


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Question Number 165385 by SLVR last updated on 31/Jan/22

$N u m b e r o f w a y s . . n$ $d i f f e r r n t t h i n g s b e d i s t r i b u t e d$ $i n r i d e n t i c a l b o x e s s o a s$ $1) e m p t y b o x i s a l l o w e d$ $2) e m p t y b o x n o t a l l o w e d$ $N u m b e r o f w a y s . . . n i d e n t i c a l$ $t h i n g s b e d i s t r i b u t e d t o r$ $i d e n t i c a l b o x e s s o a s$ $1) e m p t y b o x a l l o w e d$ $2) e m p t y b o x n o t a l l o w e d$
Commented by SLVR last updated on 31/Jan/22

$k i n d l y . . . p r o v i d e p r o o f i f$ $p o s s i b l e$
	Answered by mr W last updated on 01/Feb/22

	$I .$ $n d i f f e r e n t o b j e c t s i n t o r i d e n t i c a l$ $b o x e s$ $a) n o n e m p t y b o x$ $S (n, r) \leftarrow s t i r l i n g n u m b e r o f t h e s e c o n d k i n d$ $b) e m p t y b o x a l l o w e d$ $S (n, 1) + S (n, 2) + . . . + S (n, r)$ $I I .$ $n i d e n t i c a l o b j e c t s i n t o r i d e n t i c a l$ $a) n o n e m p t y b o x$ $p (n, r) \leftarrow n u m b e r o f p a r t i o n s o f n i n t o r p a r t s$ $b) e m p t y b o x a l l o w e d$ $p (n, 1) + p (n, 2) + . . . + p (n, r)$
	Commented by SLVR last updated on 02/Feb/22

	$S o k i n d o f y o u s i r . . . w e a r e b l e s s e d$ $H e r e i a m n e w t o S (n, r) a n d$ $i t r i e d w i t h G o o g l e s e a r c h . .$ $b u t i c o u l d n o t g e t t h r o u g h . .$ $k i d l y e x p l a i n S (9, 5) a n d S (5, 3)$ $a l s o p (n, r) w i t h p (5, 3) . K i n d l y$ $b l e s s m e s i r . . .$
	Commented by mr W last updated on 02/Feb/22
	https://en.m.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind
	Commented by mr W last updated on 02/Feb/22
	https://en.m.wikipedia.org/wiki/Partition_(number_theory)
	Commented by SLVR last updated on 06/Feb/22

	$S o k i n d o f y o u p r o f . W . . . G o d b l e s s y o u$
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