Each student chooses at least one from the \(n\) different modules offered at a university. Let us number these modules as \(1,2,3,\dots,n\). For each natural number \(1\leq k\leq n\), we denote the number of students choosing the modules \(i_1,\dots,i_k\) by \(S(i_1,\dots,i_k)\). Give a formula for the number of students in terms of the numbers \(S(i_1,\dots,i_k)\).
As an example, if \(n = 5\), \(k=3\) and we look at \(i_1 = 4,i_2 = 2, i_3 =1\), then \(S(i_1,i_2,i_3) = S(4,2,1)\) is the number of students picking the modules \(1,2,4\).