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.)

Question

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.)

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The Answer of 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.)

Let S1,S2…,S101S _ { 1 } , S _ { 2 } \ldots , S _ { 101 }S1​,S2​…,S101​

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