Did i prove the Binomial Theorem correctly? I got a feeling I did, but need another set of eyes to look over my work. Not really much of a question, sorry.
Binomial Theorem
$$(x+y)^{n}=\sum_{k=0}^{n}{{n}\choose{k}}x^{n-k}y^{k}$$
Base Case: $n=0$
$$(x+y)^{0}=1={{0}\choose{0}}x^{0-0}y^{0}=\sum_{k=0}^{0}{{0}\choose{k}}x^{0-k}y^{k}$$
Induction Hypothesis
$$(x+y)^{n}=\sum_{k=0}^{n}{{n}\choose{k}}x^{n-k}y^{k}$$
Induction Step
$$\begin{align*} (x+y)^{n+1} &= (x+y)(x+y)^{n} \\ &= x\sum_{k=0}^{n}{{n}\choose{k}}x^{n-k}y^{k}+y\sum_{k=0}^{n}{{n}\choose{k}}x^{n-k}y^{k} \\ &= \sum_{k=0}^{n}{{n}\choose{k}}x^{n+1-k}y^{k}+\sum_{k=0}^{n}{{n}\choose{k}}x^{n-k}y^{k+1} \\ &= {n\choose{0}}x^{n+1}+\sum_{k=1}^{n}{{n}\choose{k}}x^{n+1-k}y^{k}+{n\choose{n}}y^{n+1}+\sum_{k=0}^{n-1}{{n}\choose{k}}x^{n-k}y^{k+1} \\ &= x^{n+1}+y^{n+1}+\sum_{k=1}^{n}{{n}\choose{k}}x^{n+1-k}y^{k}+\sum_{k=0}^{n-1}{{n}\choose{k}}x^{n-k}y^{k+1} \\ &= {{n+1}\choose{0}}x^{n+1}+{{n+1}\choose{n+1}}y^{n+1}+\sum_{k=1}^{n}{{n}\choose{k}}x^{n+1-k}y^{k}+\sum_{k=1}^{n}{{n}\choose{k-1}}x^{n+1-k}y^{k} \\ &= {{n+1}\choose{0}}x^{n+1}+{{n+1}\choose{n+1}}y^{n+1}+\sum_{k=1}^{n}\left({{n}\choose{k}}+{{n}\choose{k-1}}\right)x^{n+1-k}y^{k} \\ &= {{n+1}\choose{0}}x^{n+1}+{{n+1}\choose{n+1}}y^{n+1}+\sum_{k=1}^{n}{{n+1}\choose{k}}x^{n+1-k}y^{k} \\ &= \sum_{k=0}^{n+1}{{n+1}\choose{k}}x^{n+1-k}y^{k}\end{align*}$$