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Num. of words of a language in func. of an alphabet, their length and a property

Source: Romanian National Olympiad, grade x, p. 4

August 25, 2019
combinatoricscountingdiscrete maths

Problem Statement

In order to study a certain ancient language, some researchers formatted its discovered words into expressions formed by concatenating letters from an alphabet containing only two letters. Along the study, they noticed that any two distinct words whose formatted expressions have an equal number of letters, greater than 2, 2, differ by at least three letters. Prove that if their observation holds indeed, then the number of formatted expressions that have n≥3 n\ge 3 letters is at most [2nn+1]. \left[ \frac{2^n}{n+1} \right] .
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