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Default COUNTIF and multiple criteria

XL2003. I've checked many of the answers in this forum, but none has worked
for me.
One column in my table has staff initials where the same few people are
scattered down the column. A dozen more columns have events marked with an X.
I want to count the number of Xs for each person. I've named the dozen
columns 'Xarray'.

I've tried variations on COUNTIF($B$2:$B$500="AB",Xarray,"X"), sometimes
CSE-ing them, sometimes using AND() and SUMIF() in different ways, but get
either an error-in-formula message, or the total number of Xs, or the total
number of cells, or FALSE or #N/A.

Please would someone tell me where I'm going wrong?
 
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