<s>[INST] Sum of the series $\binom{n}{0}-\binom{n-1}{1}+\binom{n-2}{2}-\binom{n-3}{3}+..........$ 
The sum of the series $$\binom{n}{0}-\binom{n-1}{1}+\binom{n-2}{2}-\binom{n-3}{3}+..........$$

$\bf{My\; Try::}$ We can write it as $\displaystyle \binom{n}{0} = $ Coefficient of $x^0$ in $(1+x)^n$ 
Similarly $\displaystyle \binom{n-1}{1} = $ Coefficient of $x^1$ in $(1+x)^{n-1}$
Similarly $\displaystyle \binom{n-2}{2} = $ Coefficient of $x^2$ in $(1+x)^{n-2}$
Now, how can I solve it after that, Help Required, Thanks  
 [/INST] Hint:
$$\binom{n}{0}= \mbox{ coefficient of } x^n \mbox{ in } (1+x)^n \\
\binom{n-1}{1}= \mbox{ coefficient of } x^n \mbox{ in } x^2(1+x)^{n-1} \\
\binom{n-2}{2}= \mbox{ coefficient of } x^n \mbox{ in } x^4(1+x)^{n-2} \\
...$$
Hint 2:
$$(1+x)^n-x^2(1+x)^{n-1}+x^4(1+x)^{n-2}-..=(1+x)^n \left(1-(\frac{x^2}{1+x})+ (\frac{x^2}{1+x})^2-(\frac{x^2}{1+x})^3 ...\right)$$
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