I am working on solving an equation system, that involves a parameter $N=2,3,\cdots,\infty$. Because I found it rather hard to solve it with an arbitrary $N$, I solved it in smaller numbers. One can clearly see a pattern going on:
$$\newcommand\T{\Rule{0pt}{1em}{.3em}} \begin{array}{ll} N=2 & x=1-c_1c_2 \\ N=3 & x=1-(c_1c_2+c_1c_3+c_2c_3)+2(c_1c_2c_3)\\ N=4 & x=1-(c_1c_2+c_1c_3+c_1c_4+c_2c_3+c_2c_4+c_3c_4)+2(c_1c_2c_3+c_1c_2c_4+c_1c_3c_4+c_2c_3c_4) \\ & \hspace{2.5em}-3(c_1c_2c_3c_4)\\ N & x=? \end{array}$$
Clearly, as $N$ goes up, we have some kind of "all subsets" of $c_i$: First all combinations of 2 $c_i$'s, then of 3 $c_i$'s up until all $N$ $c_i$'s.
How do I express that in general? I was thinking about something like a sum, where I allow / disallow certain indexes, but I am a bit confused..
Thank you very much!
Cheers!