Question 12
Let $k$ and $n$ be positive integers with $n\gt k$. By a binary word of length $n$ we mean a sequence of $n$ terms, all of which are 0 or 1. Anja chooses one of all possible binary words of length $n$. Then she writes on a board all binary words of length $n$ that differ from her chosen word in exactly $k$ positions. Afterwards Bernhard enters the room. Anja tells him the value of $k$, and then he looks at the binary words on the board. He now tries to guess the binary word originally chosen by Anja. What is (as a function of $k$ and $n$) the minimal number of attempts Bernhard needs in order to guess the binary word with certainty?