Algorithm for this problem on generating all permutations
Suppose that there are $n$ groups, that group $i$ has the $m_i$ elements $x_{ij}$. It is a nice exercise to show that the sum of weights of all valid vectors is
$$ \prod_{i=1}^n \sum_{j=1}^{m_i} x_{ij}. $$
(In you example, you can factor your formula as (Member 1 + Member 2) * Member 3.)