Question: A board of directors consists of $30$ members. Of them $4$ members are nominated for the position of chairperson. Each member must cast one anonymous vote. Use a generating function to determine the number of possible election outcomes.
Answer: Coefficient of $x^{30}$ in $(1 + x + x^2 + \ldots x^{30})^4.$
Here's my initial interpretation of what the coefficient of $x^{30}$ models.
Let $a, b, c, d$ be arbitrarily chosen chairpeople. Then one possible term in $(1 + a + a^2 + \ldots + a^{30})(1 + b + b^2 + \ldots b^{30})(1 + c + c^2 + \cdots + c^{30})(1 + d + d^2 + \cdots + d^{30})$ is $a^5b^7c^{10}d^{8}$ which models the situation where the candidate $a$ gets five votes, candidate $b$ gets seven votes and the candidates $c, d$ get ten and eight votes, respectively. That means the rest of the candidates get zero votes. The situation where $26$ people get zero votes happens very often in this model. Seems unrealistic. What's the more appropriate interpretation of the coefficient of $x^{30}?$ Thanks.