How do you apply the =SUM(COUNTIFS(DECENDANTS()) functions with 3 or more criteria?

I am using the =SUM(COUNTIFS(DECENDANTS()) functions and have successfully applied it for 1 criteria set with the following..
=SUM(COUNTIFS(DESCENDANTS([Contract Type Code]764), "TCH", DESCENDANTS([Instructor Type Lookup]764), "Staff"))
and also successfully applied it for 2 criteria set with the following...
=SUM(COUNTIFS(DESCENDANTS([Contract Type Code]764), "TCH", DESCENDANTS([Instructor Type Lookup]764), "FT2"), (COUNTIFS(DESCENDANTS([Contract Type Code]764), "TCH", DESCENDANTS([Instructor Type Lookup]764), "IC")))
but have not been able to replicate this formula when 3 or more criteria is required to produce the information i need.
How do I apply these functions with 3 or more criteria when using COUNTIFS?
Answers
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What is the desired result? Are you trying to add multiple COUNTIFS together? Are you getting an error or an unexpected result?
Right off I would say try this and see if it corrects the problem. If it does not, I would need more detail as to exactly what you are trying to accomplish.
=SUM(COUNTIFS(DESCENDANTS([Contract Type Code]764), "TCH", DESCENDANTS([Instructor Type Lookup]764), "FT2"), COUNTIFS(DESCENDANTS([Contract Type Code]764), "TCH", DESCENDANTS([Instructor Type Lookup]764), "IC"))
EDIT:
Give this a try first...
=COUNTIFS(DESCENDANTS([Contract Type Code]764), "TCH", DESCENDANTS([Instructor Type Lookup]764), OR(@cell = "FT2", @cell = "IC"))
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