There is an important difference between using "Can I?" and "May I?" Let's take a couple of examples:
Suppose I ask the teacher, "Can I leave the room". The teacher will answer, "Yes, you can, but you may not leave the room".
Thus when we ask "Can I?" it means we are asking if we are physically able to do something.
However, when we ask, "May I?", it means that we are asking permission.
You can remember it like this:
Can I? - Am I able to (do something)?
May I? - Do I have permission to (do something)?
Tongchan -582
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