In how many ways can 30 students be distributed to 10 schools, if 1. each school should get at least one student. 2. no restriction


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Question Number 158529 by mr W last updated on 11/Nov/21

$I n h o w m a n y w a y s c a n 30 s t u d e n t s$ $b e d i s t r i b u t e d t o 10 s c h o o l s, i f$ $1 . e a c h s c h o o l s h o u l d g e t a t l e a s t o n e$ $s t u d e n t .$ $2 . n o r e s t r i c t i o n$
	Answered by cortano last updated on 06/Nov/21

	$\Rightarrow x_{1} + x_{2} + . . . + x_{10} = 30; x_{i} ⩾ 1$ $\Rightarrow (\begin{matrix} 30 - 1 \\ 10 - 1 \end{matrix}) = (\begin{matrix} 29 \\ 9 \end{matrix}) = \frac{29!}{9! 20!}$
	Commented by mr W last updated on 07/Nov/21

	$t h i s i s o n l y t r u e, w h e n 30 (i d e n t i c a l)$ $c o m p u t e r s a r e d i s t r i b u t e d t o 10$ $s c h o o l s . b u t h e r e t h e 30 s t u d e n t s$ $a r e d i f f e r e n t o b j e c t s, s o t h e a n s w e r$ $i s n o t c o r r e c t .$
	Commented by ajfour last updated on 06/Nov/21

	$i h a d g o t$ $(1) 10!^{30} C_{10} \times 10^{20}$ $(2) 10^{30}$
	Commented by mr W last updated on 06/Nov/21

	$t h e r e s u l t f o r (1) s e e m s t o b e l o g i c,$ $b u t i t i s i n f a c t n o t c o r r e c t, b e c a u s e$ $i t i s l a r g e r t h a n t h e a n s w e r t o (2),$ $a n d t h i s c a n n o t b e t r u e, s i n c e$ $t h e w a y s w i t h r e s t r i c t i o n m u s t b e$ $l e s s t h a n t h e w a y s w i t h o u t r e s t r i c t i o n .$ $s o w h e r e i s t h e e r r o r ?$
	Commented by mr W last updated on 11/Nov/21

	$c o r r e c t s o l u t i o n s e e Q 158955$
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