A Limit Theorem from Information Theory

How many distinct sequences can be created by rearranging binary symbols? If the binary symbols are, say, and , and if the fraction of 's is denoted , then the number of possible sequences is . When is zero or one there is only one possible sequence for any value of , but when , the number of possible sequences increases exponentially with . The logarithm of the number of possible sequences, expressed on a per symbol basis, is , and the limit can be interpreted as the average number of bits needed per symbol to describe a long binary sequence with symbol probabilities and . This Demonstration shows for with the limit .

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