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Sorry for confusion, let's see if I can clarify more.

Unmarried parents share two biological children in the home, out of 3.  3rd child is mother's biological child only.  Mother signed Form 8332 until 2022, so that child's father (not in the home) can claim child as dependent.  For that child, mother wants EIC only.

Mother hasn't filed since 2008, as stay-at-home mother, but worked all of 2017, so this year is different.

This is why we want to look at creative options to maximize refund.  Father can only claim 2 of the 3 children here, but it seems to add up to much higher return, if Father claims just once, and uses mother's refund to pay what's owed.