try this where a1 contains any date in the month desired
=SUMPRODUCT(--(WEEKDAY(A1-1+ROW($A$1:INDEX(A:A,DAY(EOMONTH(A1,0)))))=7))
or start date in a1 and end date in a2
=SUMPRODUCT(--(WEEKDAY(ROW(INDIRECT(a1&":"&a2)))=7))
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