Just put each one in there. If these are all AND (they must all be
true), then keep using the * to separate each relationship. If any are
an OR, use + to separate.

=SUMPRODUCT((ADMFIN="MG")*(ADMFIN="M")*(ADMHSV<"S WG")*(ADMHSV<"NUR")*(ADMDAY=E
$3)*(ISNUMBER(MATCH(ADMFIN,{"F";"H";"N"},0))))