# Question

A certain communication system transmits text messages by representing each character with an - bit binary codeword. Suppose it is necessary for this communication system to operate in such a way that there are always an equal number of 0s and 1s transmitted. Toward that end, the communication system uses a codebook consisting only of those - bit words that have exactly n / 2 0s and n / 2 1s (where is an even integer). For example, in the case of, there are exactly 6 four- bit code words consisting of exactly two 1s and two 0s resulting in the codebook {(110), (1010), (1001), (0110), (0101), (0011)}. Thus, with four bit code words, we could represent an alphabet of only six characters.
(a) Find an expression for the number of code words with half 1s and half 0s for an arbitrary even integer.
(b) What is the minimum length of codeword we would need if the codebook needs to represent at least 100 different characters?

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