Algebraic identity for summing all arrangements of n choose k terms

Question

As the title suggests, I am looking for this type of identity, where k is some integer between 3 and n-2. I know, for example, that $$\frac{(x_1^2+x_2^2+...+x_n^2)-x_1^2-x_2^2-...-x_n^2}{2}=x_1x_2+x_1x_3...+x_1x_n+x_2x_3+...+x_2x_n +...+x_{n-1}x_n$$ Where here we have n different terms and we summed all possible ways of choosing two of those n terms. What I want to find is some identity similar to this that will sum all possible ways of choosing 3 of those n terms, or 4 of the n, or k of the n, where k is somewhere between 3 and n-2. I have found it easier to start with a smaller number of terms (say 4 instead of n), to get rid of the numerous subscripts, but I am still at a loss on how such an identity in general could be constructed. Any suggestions or ideas would be greatly appreciated!

Answer 1

This is not really an identity but is more of a property that you can utilise to find what you want. Observe the pattern among coefficients of a polynomial- $$(x-a)(x-b)(x-c)(x-d)=x^4-(a+b+c+d)x^3+(ab+ac+ad+bc+bd+cd)x^2-(abc+bcd+cda+dab)x+abcd$$ You can see that the cofficient of the $(n+1)_{th}$ term gives the sum of the roots taken $n$ at a time.