Answer: 14 cards
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Step-by-step explanation:
I'm assuming your teacher meant to say "same number or letter" instead of "same number of letter" at the very end of the problem.
The best case scenario is that the person selecting the cards is very lucky to select say a king of hearts and king of clubs. This shows that 2 cards is the lowest amount of cards to select here.
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Of course, such a scenario is not that likely. That person is likely to get some other card compared to the 1st selection.
Let's assume the person selecting the cards has the worst luck possible. This means they select each card 1,2,3,...,10,J,Q,K exactly once in any order. If this happens, then the person does not have any pairs just yet. The person has selected 13 cards so far with this worst case scenario.
The 14th card is guaranteed to pair up with some other card previously chosen. There simply aren't any other options left once you've exhausted all possibilities. See the pigeonhole principle for more information.
So this is why you would need to select 14 cards to guarantee having at least two pair up together (i.e. you would have 2 or more pairs).