Sums, Binomial Coefficient

Question

Trying this practice test and haven't got the slightest clue, any help would be great. Are professor posts the answers a few hours before the test but I'm in a lecture then

Answer 1

From the binomial theorem: $$ (1 + x)^n = \sum_{0 \le k \le n} \binom{n}{k} x^k $$ Thus: $$ (1 + 1)^n = 2^n = \sum_{0 \le k \le n} \binom{n}{k} $$

Answer 2

What is $(1+x)^n$? It is $$\sum^n_{k=0}\begin{pmatrix}n\\ k\end{pmatrix}x^k$$ Let $x=1$ to get $\sum^n_{k=0}\begin{pmatrix}n\\ k\end{pmatrix}$, thus giving $$\sum^n_{k=0}\begin{pmatrix}n\\ k\end{pmatrix}=2^n$$