Elements of Combinatorial Analysis

[Dhamnekar Win,odd]

How to use the given math hint to answer this question? 

[Dhamnekar Win,odd]

 Some more information :

 

 

 

This problem refers to the classical occupancy problem (Boltzmann-Maxwell statistics): that is, r balls are distributed among n cells and each of the [math] n^{r} [/math] possible distributions has probability [math]n^{-r}[/math]

Edited March 4, 2022 by Dhamnekar Win,odd

[Dhamnekar Win,odd]

Corrected equation (1) [math] A(r, n+1)= \displaystyle\sum_{k=1}^{r} \binom{r}{k} A(r-k, n)[/math]

Corrected equation (2) [math] A(r, n) = \displaystyle\sum_{v=0}^{n} (-1)^v\binom{n}{v}(n-v)^r[/math]

 

Then the author W. Feller says to replace in the second sum v + 1 by a new index of summation and use important property of binomial theorem which I wrote in my original question 

Edited March 6, 2022 by Dhamnekar Win,odd

[Dhamnekar Win,odd]

I tag this question as "SOLVED"

Edited March 11, 2022 by Dhamnekar Win,odd