Countif with multiple AND/OR criteria

In B4

=SUM(COUNTIFS(Accounts!B:B,A4,Accounts!F:F,"Medcare Company",Accounts!I:I,"Yes"),COUNTIFS(Accounts!B:B,A4,Accounts!F:F,"Inv Lab",Accounts!I:I,"Yes"),COUNTIFS(Accounts!B:B,A4,Accounts!F:F,"Physicians",Accounts!I:I,"Yes"))

You indicate the correct answer should be 34 but I only see 5 by looking at it and the above formula confirms 5.

I'm sorry, quick correction to the requirement here... I need a count of these account types PLUS all accounts with extended hours (regardless of account type).

Thanks!

I don't follow what exactly you need but you should be able to adapt the formula I gave you.

I think this works:
=SUM(COUNTIFS(Accounts!B:B,A4,Accounts!F:F,"Medcare Company",Accounts!I:I,"Yes"),COUNTIFS(Accounts!B:B,A4,Accounts!F:F,"Inv Lab",Accounts!$I:$I,"Yes"),COUNTIFS(Accounts!B:B,A4,Accounts!F:F,"Physicians",Accounts!I:I,"Yes"))+COUNTIF(Accounts!I:I,"Yes")

I get 33 but I counted up the extended hours and it looks like just 28 of them plus the 5 specific carriers wihch gives me only 33.

I tried, Hope this helps.

Thanks for your help. I entered the following formula which works...sorta. The problem is, the count isn't accurate because a physician with extended hours ends up being counted twice, once for being a physician and once for having extended hours. Andrew ends up with 39 instead of 34. I'm still working on it.

=SUM(COUNTIFS(Accounts!B:B,A4,Accounts!F:F,"Medcare Company"),COUNTIFS(Accounts!B:B,A4,Accounts!F:F,"Inv Lab"),COUNTIFS(Accounts!B:B,A4,Accounts!F:F,"Physicians"),COUNTIFS(Accounts!B:B,A4,Accounts!I:I,"Yes"))

the count isn't accurate because a physician with extended hours ends up being counted twice, once for being a physician and once for having extended hours

That's why I said I didn't follow exactly what you want. I assumed you didn't want a double count but I don't get what it is that you do want.
Andrew appears 44 times. He is associated with the 3 specified accounts 34 times. Only 5 of those associations have a Yes in Extended Hours.
So what is it that you are looking for?

Cutter, pretty much any of these formulas would work, except I also need the OR statement to include the Extended Hours. The original spreadsheet has 100K+ records and many of them have extended hours (I:I="Yes") but do not fall into the three given categories in F:F, hence the OR statement.

@daddylonglegs... Perfect, THANKS!!!