TITLE: Show that if seven integers are selected from the ﬁrst 10 positive integers
QUESTION [6 upvotes]: a) Show that if seven integers are selected from the ﬁrst
10 positive integers, there must be at least two pairs
of these integers with the sum 11.
b) Is the conclusion in part (a) true if six integers are
selected rather than seven?
I don't know how should I show that.
I know that in the worst situation we choose 1,2,...,7 that we have 7 + 4 = 11 and 5 + 6 = 11 but I don't know what should be the answer of this question

REPLY [6 votes]: Partition numbers from $1$ to $10$ into the following sets: $\{1,10\}$, $\{2,9\}$, $\{3,8\}$, $\{4,7\}$ and $\{5,6\}$. Now use pigeon hole principle.