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Combinations of integers

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Show that if 7 integers are selected from the first 10 positive integers there must be at least 2 pairs of these integers with the sum 11

Is the conclusion true if 6 integers are selected instead of 7

How many numbers must be selected from the set {1,3,5,7,9,11,13,15} to guarantee that at least one pair of these numbers add up to 16.

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Solution Summary

This solution discusses combinations of integers.

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The first 10 positive integers contain 5 pairs with the sum 11:
1+10 = 2+9 = 3+8 = 4+7 = 5+6 = 11
Taking 7 integers out of the 10, we leave 3 out
The ones ...

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