Graduate Fermi-Dirac statistics, finding all electron configurations

[sigint00]

Hello everyone. I'm having trouble understanding this example: https://ecee.colorado.edu/~bart/book/book/chapter2/ch2_5.htm#2_5_2

In this system of 20 electrons with equidistant energy levels, how is it known that there are only 24 possible configurations, and how are those configurations found?

 

[mfb]

By counting. You can set up fancy sums, but ultimately it is just a matter of counting.
If you want a systematic approach, sort them by energy of the highest occupied level for example, then by occupancy of that level, then by energy of the second highest occupied level, ...
Alternatively sort by number of excited electrons.

 

[sigint00]

By counting. You can set up fancy sums, but ultimately it is just a matter of counting.
If you want a systematic approach, sort them by energy of the highest occupied level for example, then by occupancy of that level, then by energy of the second highest occupied level, ...
Alternatively sort by number of excited electrons.

So there's no getting around this just being a long process of trial-and-error? or I'm misunderstanding, maybe...

 

[mfb]

24 states, it is not that long, and a computer does it in less than a millisecond.
I'm not aware of a method that is faster than counting the states, but that doesn't mean there can't be such a method.