Is there any way to abstract all unique values of sum after summation?

Question

Suppose we have 'n' numbers take 4 for simplicity like 1,2,6,7 and there sum is 'S=16'.So, Is there any way to find all unique values from 'S' using only 'n'?thank in advance..

Answer 1

My interpretation of the question:

Given $a_1 + a_2 + \ldots +a_n = S$, can we find the values of $a_i$ $(i=1,\ldots,n)$?

The short answer is no if $n \ge 2$. Take $n = 2$ for example:

$$x + y = S.$$

Then there are infinitely many choices $(x,y)=(x,S-x)$ which satisfy this equation. It's basically the equation of a line in the $x$-$y$ plane.