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

question 145

Essay

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

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Definitions:

Farsightedness

The ability to anticipate and consider future consequences and trends in decision-making or planning, often associated with strategic thinking.

Technological Synergy

The phenomenon where the interaction or cooperation of two or more technologies produce a combined effect greater than the sum of their separate effects.

Strategic Management

The process of setting goals, conducting analyses, making decisions, and implementing strategies in an organization to achieve its long-term objectives.

Scenario Analysis

A process of analyzing possible future events by considering alternative possible outcomes or scenarios.