Show that we cannot find 171 binary sequences (sequences of 0’s and 1’s), each of length 12 such that any two of them differ in at least four positions.
I assume $S$ to be any set of binary sequences of length 12 any two of them differ in at least four positions. I was just told that the answer could be found from $|S|((12C0)+(12C1)+\frac16 12C2))≤$ $2^{12 }$. But I don't know how to get this inequality.