This is a question from Shcaum's whose answer I don't understand. Our textbook has 2 pages on the pigeonhole principle and I'm having quite a bit of difficulty with it.
Give the set ${1,2,...,9}$ find how many members must be chosen to guarantee that at least one pair has a difference of 5.
There are $\binom{9}{2}=36$ possible pairs, of which exactly 4 $ \{ \{9,4\},\{8,3\},\{7,2\},\{6,1\} \} $ result in a difference of 5. They then add $\{5\}$ to the set and say there are 5 pigeonholes which requires picking a minimum of 6 numbers.
Could someone explain this?