Formula for counting number of unique values in multiple non-adjacent columns with blanks

assuming that your data ranges are A1:A10 and E1:E10, try in F1 as an ARRAY-entered formula:

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this solution will only work if you have numerical entries in your ranges.

the {1,0} trick is courtesy daddylonglegs.

UPDATE

another, simpler, more extensible solution (again, ARRAY-entered):

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in case you have other columns, such as C1:C10, the above formula can be extended thus:

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in case your data were along rows (instead of columns), the formula would be (observe the use of ";" instead of ","):

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