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I have an array of 24 batteries in my off-grid solar power system. They are arranged in a series of 4 blocks having 6 parallel-connected batteries in each. So, in my case, N=6 and m=4.

The batteries slightly vary by state of health, which is reflected in their internal resistance. This causes each of the 4 serial blocks to have slightly different resistance, which in turn causes the blocks to get charged to unequal voltage which is not ideal (some of the 4 blocks get charged to a higher voltage than the other, although they all get the same current). Given that I can measure internal resistance of each battery individually, I am about to come up with the best possible combination where the 4 blocks have the closest resistance possible.

Now, calculating resistance for a given combination is easy and is not a question.

The question is how to get all those (unordered) combinations? How many are there, and how to iterate over all of them? (so that I could write a script that would just try them all and return the best combination)

Say if N was 2 and m was 2, the 4 batteries were labeled from A to D, there would have been only 3 unordered combinations: AB+CD, AC+BD and AD+BC.

For N=3 and m=2 we get 10 (unless I have missed some):

  1. ABC+DEF
  2. ABD+CEF
  3. ABE+CDF
  4. ABF+CDE
  5. ACD+BEF
  6. ACE+BDF
  7. ACF+BDE
  8. ADE+BCF
  9. ADF+BCE
  10. AEF+BCD

So, where m=1 there is only one unordered combination no matter how big N is. But what is the equation?

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There are $24 \choose 6$ ways to choose the first pack, $18 \choose 6$ ways to choose the second, $12 \choose 6$ to choose the third, and $6 \choose 6$ ways to choose the last. We have overcounted by a factor $4!$ because we could choose the same four packs in any order. This gives $${24 \choose 6}{18 \choose 6}{12 \choose 6}{6 \choose 6}\frac 1{4!}=\frac {24!}{6!^4\cdot 4!}\approx 9.62\cdot 10^{10}$$ Generally is is $$\frac{(mN)!}{(N!)^m\cdot m!}$$

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  • $\begingroup$ Thank you! How do you call that construct with two numbers one above other in brackets? $\endgroup$ – Greendrake Apr 21 at 14:09
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    $\begingroup$ $N \choose m$ is the number of combinations of $m$ things out of $N$ without regard to order. It equals $\frac {N!}{m!(N-m)!}$ $\endgroup$ – Ross Millikan Apr 21 at 14:11

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