Why does there have to be a disjoint case in this "at least" combination problem

Question

In the problem "There are 12 fuses, 5 of which are blown-out. In how many ways can 4 fuses be selected such that at least 3 are blown-out?" The answer is

(5C3)(7C1) + (5C4)(7C0)

My question is why it cannot be

(5C3)(9)

when the nature of the remaining fuse does not matter.

Answer 1

In at least case we include given cases and above cases also.

So here are two cases when we picked 4 -

1.) Out of 5 we have picked 3 blown and out of 7 we have picked 1 is working.

2.) Out of 5 we have picked 4 blown.