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Avoiding all subsequences of length $k$

Source: Iranian third round midterm Combinatorics exam problem 2

August 27, 2019
combinatorics

Problem Statement

Let n,kn,k be positive integers so that n≥kn \ge k.Find the maximum number of binary sequances of length nn so that fixing any arbitary kk bits they do not produce all binary sequances of length kk.For exmple if k=1k=1 we can only have one sequance otherwise they will differ in at least one bit which means that bit produces all binary sequances of length 11.
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