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Find ∑k=02014(−1)k(2014k)\displaystyle\sum_{k=0}^{2014}(-1)^k\left(\begin{matrix}2014\\k\end{matrix}\right)k=0∑2014(−1)k(2014k), where (nr)=n!(n−r)!r!\displaystyle\left(\begin{matrix}n\\r\end{matrix}\right)=\frac{n!}{(n-r)!r!}(nr)=(n−r)!r!n! denotes combination.
For those of you who do not comprehend ∑\sum∑, find (20140)−(20141)+(20142)−(20143)+⋯−(20142013)+(20142014)\left(\begin{matrix}2014\\0\end{matrix}\right)-\left(\begin{matrix}2014\\1\end{matrix}\right)+\left(\begin{matrix}2014\\2\end{matrix}\right)-\left(\begin{matrix}2014\\3\end{matrix}\right)+\cdots-\left(\begin{matrix}2014\\2013\end{matrix}\right)+\left(\begin{matrix}2014\\2014\end{matrix}\right)(20140)−(20141)+(20142)−(20143)+⋯−(20142013)+(20142014)
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