80. Find: a) $C_{n}^{0}+C_{n}^{2}+C_{n}^{4}+C_{n}^{6}+\ldots ;$ b) $C_{n}^{1}+C_{n}^{3}+C_{n}^{5}+\ldots$
80. If in the identity

$$
\begin{aligned}
& (1+x)^{n}=C_{n}^{0}+C_{n}^{1} x+C_{n}^{2} x^{2}+\ldots+ \\
& \quad+C_{n}^{n-1} x^{n-1}+C_{n}^{n} x^{n}
\end{aligned}
$$

we set $x=1$, we get

$$
2^{n}=C_{n}^{0}+C_{n}^{1}+C_{n}^{2}+\ldots+C_{n}^{n-1}+C_{n}^{n}
$$

For $x=-1$ we get

$$
0=C_{n}^{0}-C_{n}^{1}+C_{n}^{2}+\ldots+(-1)^{n} C_{n}^{n}
$$

By adding and subtracting these equations term by term, we obtain

$C_{n}^{0}+C_{n}^{2}+C_{n}^{4}+\ldots=C_{n}^{1}+C_{n}^{3}+C_{n}^{5}+\ldots=2^{n-1}$.