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Combinatorial Result about the Binomial Coefficients

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Please see the attached PDF file for the fully formatted problem.

By the Binomial Theorem
....
Therefore
....
Evaluate the integral to get
....
by a similar line of reasoning.
Since this is an analysis problem, please be sure to be rigorous, and include as much detail as possible so that I can understand. Please also state if you are making use of some fact or theorem. Thanks!

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A problem involving the binomial theorem is solved. The solution is detailed and well-presented.

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