<s>[INST] Compute ${11 \choose 1} + {11 \choose 3} + ... + {11 \choose 11}$? I did this with brute force, and got 1024.
What is the faster method of solving this?
 [/INST] Since $\binom{11}{1}=\binom{11}{10},\binom{11}{3}=\binom{11}{8}$ and so on,
$$ \binom{11}{1}+\binom{11}{3}+\ldots+\binom{11}{11} = \binom{11}{10}+\binom{11}{8}+\ldots+\binom{11}{0} $$
so both the terms are half the sum $\binom{11}{0}+\binom{11}{1}+\binom{11}{2}+\ldots+\binom{11}{11}=2^{11}$, i.e. $2^{10}=\color{red}{1024}$.
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