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China Mathematical Competition

China counting and probability

Problem

Let be the set of all those -place decimal integers which contain odd number of digit '9' in each sequence . The cardinal number of is
Solution
Define as the number of the elements in , we have On the other hand, and So
Final answer
1/2*(10^2006 - 8^2006)

Techniques

Algebraic properties of binomial coefficientsSums and products