Combinatorics (Boltzmann-Maxwell statistics)

[Win_odd Dhamnekar]

This problem refers to classical occupancy problem i-e r balls are distributed among n cells and each of the \(\displaystyle n^r\) possible distributions has probability \(\displaystyle n^{-r}\)




Then the author W. Feller says to replace in the second sum v + 1 by a new index of summation and use following important property of binomial theorem.

for any number x and for any integer r,
\(\displaystyle \binom{x}{r-1} + \binom{x}{r} = \binom{x+1}{r}\)

 

[Deleted member 4993]

This problem refers to classical occupancy problem i-e r balls are distributed among n cells and each of the \(\displaystyle n^r\) possible distributions has probability \(\displaystyle n^{-r}\)
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Then the author W. Feller says to replace in the second sum v + 1 by a new index of summation and use following important property of binomial theorem.

for any number x and for any integer r,
\(\displaystyle \binom{x}{r-1} + \binom{x}{r} = \binom{x+1}{r}\)

You state:

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Where is equation (2) ?

 

[Win_odd Dhamnekar]

You state:

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Where is equation (2) ?

Equation (2) is formula (11.6) and equation (1) is (11.5).

 

[pka]

This problem refers to classical occupancy problem i-e r balls are distributed among n cells and each of the \(\displaystyle n^r\) possible distributions has probability \(\displaystyle n^{-r}\)
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What Feller calls no cell empty is known as a surjection in most combinatorics courses.
[imath]m\ge n~,~A(m,n) = \sum\limits_{k = 0}^{n - 1} {{{( - 1)}^k}\dbinom{n}{k}{{(n - k)}^m}} [/imath]
the number of onto functions from a set of [imath]m[/imath] elements to a set of [imath]n[/imath] elements.
Looking closely at that sum, we can see how the inclusion/exclusion principle is at work upon
the functions between the two sets of which there are [imath]\bf{ n^m}[/imath] such.
I suggest that you use a textbook such as An Introduction to Mathematical Statistics by
Larsen & Marx. Their treatment of combinatorics if first class.
Sadly to say that is no true of Feller.
To answer the birthday question a computer can answer [imath]\large{\dfrac{A(1900,365)}{365^{1900}}}[/imath]

 

[Win_odd Dhamnekar]

The number of distinguishable distributions A(r,n) in which no cell remains empty is \(\displaystyle \binom{r-1}{n-1} , r= objects , n= cells\).
A(r,n) can be written using another formula \(\displaystyle \displaystyle\sum_{k=0}^{n}(-1)^k \binom{n}{k} (n-k)^r\)

Now, suppose r=8 balls, n=6 cells

Using first binomial formula we get the answer 21 and using second summation formula we get the answer 191520.

How is that? If both formulas are equivalent, why these both the answers differs? Where I am wrong?