MySQL inner join COUNT optimization

Question

select COUNT(b.register_no)
from table1 a
         INNER JOIN table2 b ON a.register_no = b.register_no
         INNER JOIN table3 c ON a.register_no = c.register_no
    AND b.card_no = c.card_no;

The SQL statement takes 3 minutes to execute,Table1 has 2 million data,Table2 has 2.5 million data,Table3 has 4 million data,

explain:
table     type       Extra
tabel2    index      Using where; Using index
table3    ref        Using where
table1    eq_ref     Using index

Answer 1

a:  INDEX(register_no)
b:  INDEX(register_no,  card_no)
c:  INDEX(register_no,  card_no)

Please provide SHOW CREATE TABLE for each table.

Be aware that when JOINing a COUNT may be inflated. This is because the JOIN occurs before the COUNT . (learning's suggestion about EXISTS may give smaller numbers.)

Is there some reason to check that b.register_no is NOT NULL ? That is what COUNT(b.register_no) implies. Probably COUNT(*) is sufficient.