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QuantumP7

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## Homework Statement

Prove that

[tex] 1 - \dbinom{n}{1}\ + \dbinom{n}{2} \ - \dbinom{n}{3} \ + \cdots \ + (-1)^r \dbinom{n}{r} = (-1)^r \dbinom{n - 1}{r} [/tex]

by induction.

## Homework Equations

## The Attempt at a Solution

Well, I know how to solve the normal binomial sums by using the identity [tex] \dbinom{n}{r} \ = \dbinom{n - 1}{r} \ + \dbinom{n - 1}{r - 1} [/tex]. I'm just not sure what to do with the (-1) part. I can't find any example of an inductive proof using an alternating series anywhere, that does not involve the convergence test. Could anyone give me a hint as to where to go with this problem?

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