The best replacement for countifs is sumproduct.

The best alternatives to countifs are rated as following:
★★★★★  sumproduct
★★  countifs
★★  countif

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Speaking about countifs, one probably shouldn't forget about or statement.

Other search terms related to countifs and their ratings are:
★★★★★ or statement
★★★★ blank cells
★★★ or function
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