Formulating an equation for combinatorics Suppose I have a set $S$ of $n$ items, $S = \{ S_1, S_2, ..., S_n \}$. Each item $S_i$ is associated with some value $\alpha_i$. Now suppose I want to put the items in a bag, with the constraint that the bag can only take in $k$ items at a time.
I want to write out an equation that sums up the configurations, wherein a configuration is given by the product: $\prod$ $\alpha_i${$S_i$ is in the bag} $(1-\alpha_i)${$S_i$ is not in the bag}. 
For example, if $k=1$, the first group of products would be the case when $S_1$ was chosen to be put in the bag, the second group of products is the case when $S_2$ was chosen to be put in the bag, etc. 
$sum = (\alpha_1) (1- \alpha_2)(1-\alpha_3)...(1-\alpha_n) + (1-\alpha_1)(\alpha_2)(1-\alpha_3)...(1-\alpha_n) + ... + (1-\alpha_1)(1-\alpha_2)...(\alpha_n)$
For $k=2$, the first group of products could be the case when $S_1$ and $S_2$ were chosen to be put in the bag. The second group of products represents the case when $S_1$ and $S_3$ were chosen to be put in the bag, etc. This would produce a total of $n \choose 2$ sums.
$sum = (\alpha_1) (\alpha_2)(1-\alpha_3)...(1-\alpha_n) + (\alpha_1)(1-\alpha_2)(\alpha_3)...(1-\alpha_n) + ... + (1-\alpha_1)(1-\alpha_2)...(\alpha_{n-1})(\alpha_n)$
and so each product would be a configuration of $n$ items taken $k$. 
I'd like to know if there's a better/ more compact way to write this. Any insight would definitely help. Thanks!
 A: There definitely is a compact way to write the sum over all $k$'s of your products: it's just 1. This is because the products are the terms in the expansion of $\prod_i\bigl(\alpha_i+(1-\alpha_i)\bigr)=\prod_i1=1$. Likewise, if you want to restrict to adding up the configurations with exactly $k$ elements selected, then the sum would be the coefficient of $x^k$ in the product $\prod_i\bigl(\alpha_ix+(1-\alpha_i)\bigr)$, which is commonly denoted $[x^k]\prod_i\bigl(\alpha_ix+(1-\alpha_i)\bigr)$.
A: Here is a representation which uses the following notation which is quite common in combinatorics.
\begin{align*}
[n]\ \ &\qquad\qquad\qquad\text{the set }\{1,2,\ldots,n\}\text{ for }n\in\mathbb{N}\\
\binom{S}{k}&\qquad\qquad\qquad\text{the set of }k\text{- element subsets of }S
\end{align*}
See for instance the section notation in R.P. Stanleys Topics in Algebraic Combinatorics.
A formulation of OPs example using this notation could be

Let $S=\{S_1,S_2,\ldots,S_n\}$ be a set of $n$ items. We associate to each item $S_i,1\leq i\leq n$ a weight $\alpha_i$  with  $0\leq \alpha_i\leq 1$. For each $k\ (0\leq k\leq n)$ we consider $k$-size bags which contain $k$ pairwise different items $S_i$ from $S$.
For a specific $k$-size bag let the $k$-subset  $A\in \binom{[n]}{k}$ be the corresponding index set. The price of a $k$-size bag  is then given as
  \begin{align*}
\prod_{i\in A} \alpha_i \prod_{j\in [n]\setminus A}(1-\alpha_j)
\end{align*}
  The price of all possible $k$-size bags is therefore
  \begin{align*}
\sum_{A\in\binom{[n]}{k}}\prod_{i\in A} \alpha_i \prod_{j\in [n]\setminus A}(1-\alpha_j)
\end{align*}

