It essentially ignores column C's missing entry and only considers the other two. ( Although the fourth criterion in your example formula is not obvious from your list, the procedure can be expanded to include as many criteria as you may have.
=SUMPRODUCT((IF(LEN(C2),(INDEX(Lists,,1)=C2),TRUE))*(IF(LEN(C3),(INDEX(Lists,,2)=C3),TRUE))*(IF(LEN(C4),(INDEX(Lists,,3)=C4),TRUE))*SumRange)
To make things easier to comprehend, I made a named range Lists that includes your sample range E2:H10 and labelled I2:I10 as SumRange. The first column in the range is shown by INDEX(Lists,,1). It's crucial that Lists and SumRange both include the same amount of rows.