Q:

About binomial coefficient

I have a problem in the calculation above. Still haven't learned how to use the LaTex language so I uploaded it this way. I reached to a wrong answer, just tell me please what's wrong with my way of thinking or the steps I've taken.

A:

It is handsome to work under the convention that $\binom{n}{k}$ is defined for $n\in\{0,1,2,\dots\}$ and $k\in\mathbb Z$ and takes value $0$ if $k\notin\{0,1,\dots,n\}$.
Under that convention the equality: $$\binom{n-1}{k-1}+\binom{n-1}{k}=\binom{n}{k}$$ is true for $n\geq1$ and $k\in\mathbb Z$, so the first equality mentioned in your question is immediate.
Further:
$$\sum_{k=0}^n\binom{n}{k}=2^n=2\times2^{n-1}=2\times\sum_{k=0}^{n-1}\binom{n-1}{k}$$
So there is nothing wrong.
Under the convention you can also write this as:$$\sum_{k\in\mathbb Z}\binom{n}{k}=2^n=2\times2^{n-1}=2\times\sum_{k\in\mathbb Z}\binom{n-1}{k}$$

