Binomial expansion and summation notation question

Question

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?

Answer 1

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.

Answer 2

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.