Prove that if six numbers are chosen from the set 2468101214

Prove that if six numbers are chosen from the set {2,4,6,8,10,12,14,16,18,20} then at least one pair adds up to 22. (Suggestion. Use the Pigeonhole Principle.)

Solution

A={2,4,6,8,10,12,14,16,18,20}

S={ (2,20), (4,18), (6,16), (8,14), (10,12) }

Clearly, S is the set of \"pairs of numbers which add up to 22\" Every element of A belongs to one of the pairs in S

S has total 5 elements (or 5 pairs).

Now, if we choose 6 elements from A then, by pigeonhole principle, not all 6 elements can belong 5 distinct pairs in S. That is, atleast one pair from 6 chosen elements has to be in S.

Thus, there exists a pair that adds upto 22.