Determine the cycle representation of the permutation \(\left(\begin{array}{cccccccc} 1&2&3&4&5&6&7&8\newline 3&8&1&4&7&2&5&6\end{array}\right)\).
Assume that \(a_n=3\) for \(n\geq 0\). Let \(F\) be the generating function of the sequence \((a_n)_{n=0}^{\infty}\). Determine \(F\left(\frac12\right)\).
Let \(a_n\) be the number of ways to distribute \(n\) apples to \(5\) people so that all of the following 3 conditions are satisfied:
Determine the generating function for the sequence \((a_n)_{n=0}^{\infty}\). You do not need to evaluate \(a_n\).
Let \(f=(1592)(374)(68)\) and \(g=(173)(248)(596)\) be two permutations on the set \(\{1\), \(2\), \(3\), \(4\), \(5\), \(6\), \(7\), \(8\), \(9\}\).
Denote by \(a_n\) the number of sequences of length \(n\) consisting of letters from the set \(\{A,B,C\}\) such that no two letters \(A\) are consecutive.
Let \(a_n\) denote the number of ways to distribute \(n\) apples to \(10\) people such that the first person gets an even number of apples. Let \(t_n\) be the total number of ways to distribute \(n\) apples to \(10\) people.