Math question

Can someone help me with this question??
How many positive integer divisors of 2004^2004are divisible by exactly 2004 positive integers?

I understand the first part, but I don’t really get the second part about counting

Do you understand where (x+1)(y+1)(z+1) comes from?

Yeah (just until there)

From there on its just a simple diophantine/integer equation so
(x+1)(y+1)(z+1)=2004
as the lhs represents the number of distinct factors/divisors, so prime factorise the rhs (which youve already done) and then consider how many different ways you have for equating the factors of the right to the terms on the left.

What I don’t really understand is how they’ve calculated the number of different ways when equating factors - I thought it would be sth like 3*2*1 (permutations?)