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Default Rank based on 2 categories

I was using RANK() to assign places based on a score, but now I have to
separate the places by team. I found a post suggesting an array formula, but
there are some problems with it. In column A (named "Team") is the team name;
column B (named "Score") is the score upon which I want to rank, and columns
C, D, and E are the columns where I want the rank to appear. I'll post sample
data below.

Here is the formula in column C (for Team A):
{=IF(A2<"Team
A","",COUNTIF(Team,A2)+1-MATCH(TRUE,B2=LARGE(Score*(A2=Team),ROW(INDIRECT(" 1:"&COUNTIF(Team,A2)))),0))}

For columns D and E, "Team A" is changed to "Team B" and "Team C"

The problems a
1. When the score column is empty, the formula returns the number of people
on that team, instead of blank.
2. It ranks scores in reverse order (highest score should be #1, but this
formula makes the lowest score #1).
3. When there is a tie, it gives both people the lower place (for Team B
below, the ranks should be 2, 2, 1, instead of 3, 3, 1 (and if ranked in the
correct order, it would be 1, 1, 3)).

Team Score Rank A Rank B Rank C
Team A 5 3
Team A 4 2
Team A 3 1
Team B 4 3
Team B 4 3
Team B 1 1
Team C 5 3
Team C 3 2
Team C 3 2
 
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