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  1. I’ll finish the work in the next few days. You can pay me then.

  2. Although you can visit these places, you’re welcome to stay in the hotel if you are tired.

  3. You can buy a ticket from the Tourist Information Center.

Can I use "may" in place of "can" in (1), (2) and (3)?

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    Not necessarily in all of your examples, but as a general principle, may carries overtones of being granted permission, whereas can usually just implies capability. Also in many contexts, may can imply some degree of feasibly / likelihood which is rarely alluded to by can. – FumbleFingers Feb 22 at 18:54
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"May" can replace "can" in all three examples. "You may" also can be interpreted as "you are permitted". "Can" should, as a rule, describe the ability to do something.

In examples 1, and 3, "can" makes an assumption about the ability & availability of funds that might not prove true. The fact that you finish the work in a few days does not mean I can pay you then, and similarly, tickets being sold at the Tourist Information Center does not equate to my ability to purchase one.

Example 2 could be rewritten with "might" to illustrate the difference -- "Although you might do some thing, you might do some other thing instead. But you can not do both things."

Finally, your question: "Can I use...": the issue is, you can use either. No one prevents you from doing so. Can you use either without confusing your audience? Absolutely. Can you use either while also strictly adhering to the defined usage of the words 'can' and 'may' and demonstrating your understanding of their conceptual difference for the purposes of a test question? No. In all three cases, "may" will be the more correct option. "Should I use 'may' in place of 'can'?" -- yes.

Err on the side of being absolutely correct on this distinction until it's second nature, and then one may become lazier about it. You'll find that, idiomatically, 'can' substitutes for 'may' with increasing frequency, particularly in US/US-inflected English spoken usage, and with the resulting consequence that substituting 'may' in any of these examples might strike the audience as a jarring formality or, worse, condescending and passive aggressive.

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    At school I had a teacher who, if you asked "Can I go to the toilet, sir?" would invariably answer "I have no doubt that you can; the question is whether you may." – Daniel Roseman Feb 22 at 21:29

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