I am trying to understand my teacher's mathematical development.
Just to give you some context, $\vec{c}$ is a boolean vector and this is the probability of measuring $\vec{c} = (c_1,...c_n)$. When the teacher is summing over $b_1,...,b_n$, he is summing over all boolean vectors (i.e the first iteration of the sum is when $(b_1,...,b_n) = (0,...,0)$ and the last iteration of the sum is when $(b_1,...,b_n) = (1,...,1)$)
I do not understand how the teacher gets from line 2 to line 3. How does he get rid of the sum ? I tried writing the sum (term by term), but I do not get the same thing he does. Could you help me out there, please?
For those who are interested in knowing what this represents, this is the probability of measurement in the Deutsch-Jozsa quantum algorithm when the function is constant (given that my questions is about algebra, I do not think it is useful for me to specify the details of the algorithm for you to help me).
Let's go from (3) to (2). The product in (3) has the form $$(x_{1,0}+x_{1,1})(x_{2,0}+x_{2,0})\cdots(x_{n,0}+x_{n,1}).\tag{*}$$ Here $x_{i,0}=(-1)^{c_i0}$ and $x_{i,0}=(-1)^{c_i1}$.
Expanding out (*) gives $2^n$ terms, a typical one is $$x_{1,b_1}x_{2,b_2}\cdots x_{n,b_n}$$ where each $b_i$ is chosen from the set $\{0,1\}$. In your case, that gives the sum in (2).
In general we have $$\prod_{i=1}^n(x_{i,0}+x_{i,1})=\sum_{b_1=0}^1\sum_{b_2=0}^1\cdots\sum_{b_n=0}^1 \prod_{i=1}^n x_{i,b_i}.$$