I came across this question in my book today. The question asks me to proof this by rewriting $(1+x)^n$ in summation form. I can't think of any relationship between the two equations other than expanding then in $\binom{n}{r}$ form. Can you guys give me some hints? 
Binomial expansion and summation notation question
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$\begingroup$
summation
binomial-theorem
2 Answers
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We know that from the binomial theorem, $$(1+x)^n = 1 + x\binom{n}{1} + x^2\binom {n}{2} + x^3\binom {n}{3} + \cdots + x^n \binom {n}{n}.$$
Now if you let $x = 1$, the left side becomes $2^n$ and the right side is $\sum_{r=0}^{n} \binom {n}{r} = \binom {n}{0} + \binom {n}{1} + \cdots + \binom {n}{n} $, which is what you want! Hope it helps.
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By the binomial theorem, you get
$$ (1+x)^n = \binom{n}{0} + \binom{n}1 x + \cdots \binom{n}n x^n. $$
Now if you substitute $x=1$, you get your result.