Unfortunately, there aren't any cell formulas that can check for fill color. That requires either data filters or VBA and neither would be suitable in your case as I see it.

There might be a way to do what you want only it would take a bit of work and will make your workbook twice as big. What you could do is have a corresponding helper column check the validity of each of those cells and return a value, say 0=valid and 1=invalid. These cells could be hidden to make the sheet more visually manageable. Then you would just have one formula that would add up all these cells and display either a valid/invalid or even just the sum (to indicate how many invalids).

It's kind of a kludge, I know, but Excel don't care. :-)