Problem 484

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

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