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

.