mysql COUNT results with GROUP BY

Your requirement is to get count of number of groups. So we need two operations-

Group(inner query)

Count(outer query)

Following query will do precisely that:

SELECT COUNT(*)
FROM
(
    SELECT COUNT(id) 
    FROM my_table 
    GROUP BY first_linked_table_id,
             second_linked_table_id
) t

If you want to count the rows, I think you're going to need a subquery. Something like this:

SELECT COUNT(*) FROM (
    SELECT COUNT(id) FROM my_table GROUP BY first_linked_table_id, second_linked_table_id
);

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