Let $S _ { 1 } , S _ { 2 } \ldots , S _ { 101 }$

Essay

Let $S _ { 1 } , S _ { 2 } \ldots , S _ { 101 }$ be 101 bit strings of length at most 9 . Prove that there exist two strings, $8$ and $S _ { j }$ , where $i \neq j$ , that contain the same number of 0 's and the same number of 1's. (For example, strings 001001 and 101000 contain the same number of 0 's and the same number of 1 's.)