Problem
COM-B2-M09-P012 Symmetry of Coefficients
#12
★★★☆☆ Level 3 of 5
Let \(c_k\) be the coefficient of \(x^k\) in \((1+x+\cdots+x^m)^n\). Prove that \(c_k=c_{mn-k}\).
The coefficient counts tuples \((a_1,\ldots,a_n)\), where \(0\le a_i\le m\). Replace \(a_i\) by \(m-a_i\).
The coefficient \(c_k\) equals the number of tuples \((a_1,\ldots,a_n)\) such that \(0\le a_i\le m\) and \(a_1+\cdots+a_n=k\). Map such a tuple to \((m-a_1,\ldots,m-a_n)\). The new numbers still lie between \(0\) and \(m\), and their sum is \(mn-k\). This correspondence is reversible, so \(c_k=c_{mn-k}\).
Prepares students to use symmetry of bounded coefficients without expanding the polynomial.