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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 $n$.

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  • $\begingroup$ Note that: $(x+a)(x+b)(x+c) = x^3 + (a+b+c)x^2 + (ab+bc+ac)x + abc$ $\endgroup$ – anonymouse Mar 9 '16 at 15:27
  • $\begingroup$ Yeah this could be dubbed the 'polynomial coefficient'. $\endgroup$ – user321329 Mar 9 '16 at 15:29
  • $\begingroup$ Maybe I misunderstood the question: what do you mean by "mathematical expression which describes the process I undertook above"? $\endgroup$ – anonymouse Mar 9 '16 at 15:30
  • $\begingroup$ Well the main goal was to produce a quick way of expanding any factored polynomial. So if I had $(x-a)(x-b)(x-c)(x-d)(x-e)$, I can write out $x^5 - x^4 + x^3 - x^2 + x - 1$, then I can assign the coefficients using the method I did above. I just wanted a more elegant way to express it. (Think similar to how the binomial series is expressed) $\endgroup$ – user321329 Mar 9 '16 at 15:34
  • $\begingroup$ My gut feeling is that I'm skeptical that such a method exists (unless you want to research some efficient polynomial multiplication), but that feeling is just on first glance. Let me think about it some more, and maybe someone else will have better insight into this. $\endgroup$ – anonymouse Mar 9 '16 at 15:40
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If you multiply out $(1+ax)(1+bx)\cdots(1+ex)$, you’ll get $$1 + (a + b + c + d + e) x + (a b + a c + b c + a d + b d + c d + a e + b e + c e + d e) x^2 + (a b c + a b d + a c d + b c d + a b e + a c e + b c e + a d e + b d e + c d e) x^3 + (a b c d + a b c e + a b d e + a c d e + b c d e) x^4 + (a b c d e )x^5,$$ so the coefficients of $x^k$ are your $C_k$. If you want to do this for any number of elements, you can name the elements $a_i$, and then $_NC_k$ is the coefficient of $x^k$ in $\prod_{i=1}^N(1+a_ix)$. The phrase describing this sort of thing is “generating function.”

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These values are classically known as elementary symmetric polynomials in the "constants" belonging to your finite set $S = \{a,b,c,d,...\}$.

Their connection with expressing the coefficients of a polynomial in terms of "generic" roots of the polynomial have been studied at least since the time of Isaac Newton and a slightly earlier scholar, Albert Girard.

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