Q-pigeon hole: a group of 52 sequences, each 9 long from the group {0,1}

Question

let A be a group of sequences, each sequence length is 9 and each od its elements is from the group {0,1} we know that |A| =52. show that there exsits at least 2 sequence a1 and a2 such that you can switch at most 2 elemnts in a1 and get a2

Answer 1

I’ll assume that you meant “switch at most $2$ elements”, since the statement is false as written.

Around each sequence is a ball of $10$ sequences that differ from it in at most $1$ element. If there were no two sequences at distance at most $2$, these balls could not overlap. There would be $10\cdot52=520$ different sequences in these balls, but there are only $2^9=512$ different binary sequences of length $9$.