Am I correct in saying:
COUNT(expr)
WHERE expr IS NOT *
Will count only non nulls?
Will COUNT(*) always count all rows? And What if all columns are null?
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Correct. COUNT(*) is all rows in the table, COUNT(Expression) is where the expression is non-null only.
If all columns are NULL (which indicates you don’t have a primary key, so this shouldn’t happen in a normalized database) COUNT(*) still returns all of the rows inserted. Just don’t do that.
You can think of the * symbol as meaning "in the table" and not "in any column".
This is covered in the MySQL Reference Manual.