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Prove using the binomial theorem that alternately adding and subtracting elements across a row of Pascal’s triangle always results in zero.

I need help constituting a proof. I am able to show this works for specific cases and by substituting numbers into the binomial theorem but how will I go about a general proof that illustrates this works for every case?

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  • $\begingroup$ @Parcly Taxel Doesnt seem like a duplicate since OP has not found the link with $\sum_k \binom{n}{k} (-1)^k $ $\endgroup$ – krirkrirk Feb 28 '18 at 12:44
  • $\begingroup$ sorry didn't realize it was already asked. Thank you! $\endgroup$ – Lil Feb 28 '18 at 14:52
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$$ 0 = (-1+1)^n = \sum_{k=0}^n {n\choose k} (-1)^k 1^{n-k} = \sum_{k=0}^n {n\choose k} (-1)^k. $$

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