Prove without Binomial theorem

Question

For $n >= 0$

$$C(n,0)+C(n,1)+C(n,2)+...+C(n,n)=2^n$$

I know how to solve this using Binomial theorem.

But now, I am asked to find a simple combinatorial proof of it.

Answer 1

Number of subsets from a set of $n$.

Your subsets can contain $0,1,2,...,n$ members. There are $\binom{n}{i}$ subsets of size $i$. Therefore $\sum_{i=0}^{n}\binom{n}{i}$ give You the number of all possible subsets from a set of $n$.

Every element of set of $n$ maybe included or not included in our subset ($2$ possibilities). Since there are $n$ elements, there are $2^n$ possible subsets.

Therefore $\sum_{i=0}^{n}\binom{n}{i}=2^n$