I saw the binomial theorem in some 11th grade book in my library. It was proved by induction. But there was no derivation of it. I do not like to memorize it. So I'm hoping someone can show me a derivation for it and give hint for another way to derive it. It would be great if you try to use high school mathematical tools if possible, so I can understand it.
$$ (x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}=\sum _{k=0}^{n}{n \choose k}x^{k}y^{n-k}. $$