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Combinatorial Result about the Binomial Coefficients

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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Solution Summary

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

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