Question

Let A={0,1,2,3,4,5,6,7,8,9,10}.Show that If S is any subset of A with 7 elements,then there are 2 elements of S whose sum is 10?

Answer 1

Question 1.

Let $A=\{1,2,3,4,5,6,7,8,9,10\}$. Show that if $S$ is any subset of $A$ with 7 elements, then there are 2 elements of $S$ whose sum is 10.

Solution. There are 4 unordered pairs of numbers in $A$, whose sum equals 10:

$\{1,9\},\{2,8\},\{3,7\},\{4,6\}.$

The rest 2 numbers (5 and 10) do not have a corresponding complement to 10. Since $4+2=6<7$, then if we choose arbitrary 7 numbers from $A$, by Pigeonhole principle we necessarily find at least one above pair among them. $\Box$