I am lost with the signs cancellation. Please help me to calculate this inner pruduct.
Let $a$ and $b$ be two $2m$ dimentional vectors such that their entries are Rademacher random variables and such that the sum of the variables for each vector is zero. i.e. $P(a_i=1)=P(a_i=-1)=P(b_i=1)=P(b_i=-1)=\frac{1}{2}$ and $ \sum\limits_{i=0}^{2m}a_i=\sum\limits_{i=0}^{2m}b_i=0 $ Find the inner product $\langle a,b\rangle$.