Given m vectors $v_1, v_2, ...,v_m$ in $\mathbb{R}^n$ with coordinates $0$ or $1$. Each vector is a result of $n$ Bernoulli trials. It may be that $v_i = v_j, i \neq j$.
Consider three vectors: $v_i, v_j, v_k$. Say that it is a "good" triple of vectors if $ \langle v_i - v_k , v_j - v_k \rangle = 0 $
How find Expected value of number of "good" triples in the set $\{ v_1, v_2, ..., v_m \} $?
I guess I have to use characteristic functions to solve the problem?
The contribution of a coordinate to the scalar product is either $0$ or $1$, hence the scalar product is $0$ if and only if all contributions are $0$, which means that in each coordinate at least one of $v_i, v_j$ equals $v_k$. If we fix $v_i$ and $v_k$ and they agree on $r$ coordinates, then the probability for a third vector $v_j$ to make this a good triple is $2^{r-n}$. Hence each pair with $r$ coordinates agreeing contributes $2^{r-n}\cdot(m-2)$ to the expected value. The probability of $r$ coordinates agreeing is ${n\choose r}2^{-n}$ so that each pair contributes $$\sum_{r=0}^n {n\choose r}2^{-n}2^{r-n}(m-2)=2^{-2n}(1+2)^n(m-2)$$ to the expected value and after summing over all ${m\choose 2}$ possible pairs $(i,k)$ we obtain $$ E=\frac{3^nm(m-1)(m-2)}{2^{2n+1}}.$$