I could do the exam late because I was ill

[xxwzs]

It is said that we give or refuse permission with can or may, not could or might in 《Oxford English grammar course - Advanced by Michael Swan and Catherine Walter 》(Page 62), where two sentences are given for audience to make a judgement.
A. I could do the exam late because I was ill.
B. At school, we could choose classics or science, but we couldn't mix them.
Sentence A is said to be incorrect, but Sentence B correct in 《Oxford English Grammar》. Then I have got puzzled:
1. Will A be correct if 'could' is replaced with 'can'?
2. Now that 'could' is not used for giving permission, why B is correct? If 'coulds' are replaced with 'cans' in B, is B still correct?

 

[5jj]

1. Will A be correct if 'could' is replaced with 'can'?

No, you couldn't do the exam.

2. Now that 'could' is not used for giving permission, why is B is correct? If 'coulds' are replaced with 'cans' in B, is B still correct?

No. Can does not refer to past time.

At school, we could choose classics or science, but we couldn't mix them. Past time
At school, we can choose classics or science, but we can't mix them. Present/general time/

 

[jutfrank]

Sentence B isn't about having permission.