How to prove $\left(\!\!\binom{n}k\!\!\right)=\left(\!\!\binom{k+1}{n-1}\!\!\right)$ (multichoose) algebraically?

Question

How to prove algebraically that

$$\left(\!\!\binom{n}k\!\!\right)=\left(\!\!\binom{k+1}{n-1}\!\!\right)$$

I am having a good amount of trouble understanding the algebraic side of problems like these. I have the combinatorial side down, but for some reason, I really struggle with the breakdown of each side of the proof and how the canceling works.

Answer 1

Wolfram mathworld says$$\left({n\choose k}\right)={n+k-1\choose k}$$

This means your claim that $$\left({n\choose k}\right)=\left({k+1\choose n-1}\right)\\\implies {n+k-1\choose k}={k+1+n-1-1\choose n-1}$$

is Absolutely Trival!