Find the minimum number of elements that one needs to take from a set $$ S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$$ to be sure that there exists at least 2 numbers with the sum of 10.
Here are my work so far:
These are the pairs with the sum of 10: (1, 9), (2, 8), (3, 7), (4, 6)
If we select 4 numbers, then at least 2 of them must be in the same pairs above. But then the set also contains 5, so I suppose that we need at least 5 numbers to safely have a pair with a sum of 10 - is this correct?
If we take $6$ of them then there will be at least one pair giving a sum of $10$.