Suppose that I have a set of $n$ constants. I want to find the sums of all product combinations of length $i = n \rightarrow 0$
Using $n=5$ as an example:
$C = {a,b,c,d,e} $
$C_5= abcde$
$C_4= abcd+abce+abde+acde+bcde$
$C_3= abc+abd+abe+acd+ace+ade+bcd+bce+bde+cde$
$C_2= ab+ac+ad+ae+bc+bd+be+cd+ce+de$
$C_1= a+b+c+d+e$
$C_0 = 1 $
Using the binomial coefficient, I can determine the number of terms there should be for each combination length, since it corresponds to ${n \choose k}$ for $n$ constants and $k$ length. However, I am looking for a mathematical expression which describes the process I undertook above, where I produced the sum of all the unique products of $k$ terms, but for any amount of constants $n$.