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Default SUMPRODUCT with an OR component

I Have a SUMPRODUCT function which is working fine to return a count of rows
meeting about 7 different criteria (e.g., --(A1:A4000="Yes"), etc.). To this
function I need to add a component which returns TRUE if at least one of
another dozen or so conditions are TRUE. So it's like putting in another
dozen sets, all wrapped in an OR function. I tried variations of
"--(OR(B1:B4000<1,C1:C4000<0, etc.)" and then array-entering the whole
SUMPRODUCT function, but that doesn't work. Any ideas?

TIA
 
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