Knowing that : Σ_(k=0) ^n ((n),(k) )^2 = (((2n)),(( n)) ) We have p tokens and n boxes. Each box is labeled with a number from 1 to n. Each box is enough big to receive all p tokens. In how many ways can we share the token : a) if we can distinguish all tokens ? b) if all the tokens are the same ? NB : the boxes may have 0,1,2...,p tokens in them.


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Question Number 125413 by Hassen_Timol last updated on 11/Dec/20

$Knowing that :$ $\underset{k = 0}{\sum^{n}} {(\begin{matrix} n \\ k \end{matrix})}^{2} = (\begin{matrix} 2 n \\ n \end{matrix})$ $We have p tokens and n boxes .$ $Each box is labeled with a number from 1 to n .$ $Each box is enough big to receive all p tokens .$ $In how many ways can we share the token :$ $a) if we can distinguish all tokens ?$ $b) if all the tokens are the same ?$ $NB : the boxes may have 0, 1, 2 . . ., p tokens in them .$
Commented by Hassen_Timol last updated on 10/Dec/20
Could you help me please...?
	Answered by mr W last updated on 11/Dec/20

	$a)$ $n^{p}$ $b)$ ${(1 + x + x^{2} + . . .)}^{n} = \frac{1}{{(1 - x)}^{n}} = \underset{k = 0}{\sum^{\infty}} C_{k}^{k + n - 1} x^{k}$ $c o e f f i c i e n t o f x^{p} i s C_{p}^{p + n - 1}$
	Commented by Hassen_Timol last updated on 11/Dec/20
	Thank you a lot ! I am sorry that I didn't understand but what is x ?
	Commented by mr W last updated on 11/Dec/20

	$i u s e d g e n e r a t i n g f u n c t i o n m e t h o d .$ $x i s h e r e j u s t a s y m b o l f o r v a r i a b l e .$ $w e c a n a l s o s o l v e (b) u s i n g s t a r s &$ $b a r s m e t h o d u n d g e t C_{n - 1}^{p + n - 1} w h i c h$ $i s t h e s a m e a s C_{p}^{p + n - 1} .$
	Commented by Hassen_Timol last updated on 12/Dec/20
	Thank you very much
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