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Guest Message by DevFuse
 

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Colored balls in a bag


Best Answer plasmid, 11 May 2014 - 06:02 PM

Spoiler for atrocious but verified answer

 

Spoiler for this agrees with simulation - perl code
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#11 m00li

m00li

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Posted 25 May 2014 - 03:16 AM

More recursions:

 

Let f(t,m) denote the number of ways of arranging m colored balls in t slots, where t >= m and supply of each m coloured ball is infinite. Then final answer is nCmf(t,m) / nt

 

Here are the different recursions I have come up with for f(t,m)   (alas, no direct formula in m,n)

 

1) mtmC1f(t,1) - mC2f(t,2) - mC3f(t,3) .... - mCm-1f(t,m-1) 

 

2) tC1f(t-1,m-1) + tC2f(t-2,m-1) + tC3f(t-3,m-1) + ... + tCt-m+1f(m-1,m-1)

 

3) mf(t-1,m-1) + m2f(t-2,m-1) + m3f(t-3,m-1) + ... + mt-m+1f(m-1,m-1)

 

4) m( f(t-1,m) + f(t-1,m-1) )

 

where f(t,1) = 1, f(t,2) = 2t-2, and f(t,t) = t!

 

And finally one kind of non recursive form:

 

5)     f(t,m) = mtmC1(m-1)tmC2(m-2)tmC3(m-3)t + ....(-1)m-1(mCm-1)  (This one gives the AHA insight into the solution) :)


Edited by m00li, 25 May 2014 - 03:19 AM.

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