I am trying to write a generalized equation to solve a fairly simple probability problem (c & k are constants)
$$y_{1} = (1 - cx_{1})^k$$
$$y_{2} = \frac{(1 - cx_{2})^k - x_{2}[1-c[1-(1-x_{1})(1-x_{2})]]^k}{(1-x_{2})}$$
$$y_{3} = \frac{(1-x_{2})(1-cx_{3})^k - x_{3}(1-x_{2})[1-c[1-(1-x_{1})(1-x_{3})]]^k - x_{3}[1-c[1-(1-x_{2})(1-x_{3})]]^k + x_{3}[1-(1-x_{2})(1-x_{3})][1-c[1-(1-x_{1})(1-x_{2})(1-x_{3})]]^k}{[(1-x_{2})(1-x_{3})^2]}$$
If S is a set of i length combinations of $x_{1}...x_{j-1}$ such that for eg $${\{x_1,x_2,x_3,x_4\} \choose 2} = \{\{x_1,x_2\},\{x_1,x_3\},\{x_1,x_4\},\{x_2,x_3\},\{x_3,x_4\}\}$$ Then I can see that the terms to the power k for the $y_j$ term might be written along the lines....
$$\Sigma^{j-1}_{i=0}(-x_j)^{j+1-i}[1-c[1-(\Pi_{r\in{S \choose i}} (1-r) )(1-x_{j})]]^k$$
But even if that is correct (which I doubt), I still run into problems generating the other terms. I'm sure there is a solution to this but I'm not a mathematician so I'm looking for a nice way to generalize it.