One way would be to add a helper column which returns 1 for each row if any of the relevant values equal 1. Then a simple SUM formula can sum up the 1s and 0s in that helper column. The helper column formula would be something like this, copied down the column:
Another option would be to adapt your COUNTIFS formula like this:
This option would not be practical for the potential combinations of 15 COUNTIFS() functions.
Another option would be to use SUMPRODUCT.
If you go for the SUMPRODUCT option, try not to use whole column references because it will slow your workbook down.
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