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The only way to deal with this efficiently is to reduce the input sets to non-overlapping sets prior to permuting them.

To do that, you need to lay out the rules for overlapping sets. You'd need to define how to deal with:

One set completely overlaps another;
1. the overlap is less than the number to be picked from the smaller group.
2. The overlap is less than the number to be picked from the larger group.
3. The overlap is equal to the number to picked from the smaller group.
4. ...
There is only a partial overlap between the groups:
1. The overlap is smaller than the number to be picked ....
2. ...

Whence you've defined how all the possibilities should be resolved into distinct groups, the rest should be relatively easy.

I'm still bugged by your lack of explanation of what the purpose of this is?

With the rise and rise of 'Social' network sites: 'Computers are making people easier to use everyday'

Examine what is said, not who speaks -- Silence betokens consent -- Love the truth but pardon error.

"Science is about questioning the status quo. Questioning authority".

In the absence of evidence, opinion is indistinguishable from prejudice.

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