How to compile data from multiple sheets and then test that against another sheet

[Previously titled, "How to create something like:COUNTIF(A1:J10, AND(OR(Sheet1!A1=1, Sheet2!A1=1), Sheet3!=1))"]

Sorry, that title [is probably] was terrible, but I'm having trouble describing what I need to do, and that imaginary function is pretty close.

I have many sheets of binary data, and several sheet of "designed" data (also binary), all of which are the same range. I need to test how well different combinations of the data sheets match the designed data.

I can do this for any single test with a helper sheet. On the helper sheet, I create an IF statement in each cell in the range that uses an OR to check if any of the data sheets have a 1, and then uses AND to check that against the designed sheet. E.g., =IF(AND(OR(Sheet1!A1=1, Sheet2!A1=1, Sheet3!A1=1), Design!A1=1), 1, 0). And then since I know how many positive cells a sheet should contain (all the designed sheets have the same number of positive cells), I just sum the helper sheet.

The attached example workbook shows what I'm trying to accomplish for a single case, using the method I described.

The problem is that I need a way to do it at scale, since I have a few dozen data sheets that need to be tested in different combinations and a handful of designed sheets. So the total workload here is possibly a couple thousand iterations of this. The only information I need for each iteration is the sum on the helper sheet (I don't need to know which cells in each test fail to match).

Any help is greatly appreciated. I can work with VB code if that's the best way to do this. Thank you for any ideas!

Administrative Note:

Yes, the title is terrible - sorry.

We would very much like to help you with your query, however the thread title does not really convey what your request is about.

Please take a moment to amend your thread title. Make sure that the title properly explains your request. Your title should be explicit and not be generic (this includes function names used without an indication of what you are trying to achieve).

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Sorry about that, post has been re-titled.

Thank you.