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MTH 4150: Midterm 2 Practice 3

Problem 1

Let\[ \sigma=\left(1352\right)\left(46\right)\left(897\right) \quad\mbox{and}\quad \rho=\left(\begin{array}{ccccccccc}1&2&3&4&5&6&7&8&9 \newline 7 & 8 & 2 & 3 & 4 & 5 & 1& 9 &6 \end{array}\right) \] bet two permutations of the set \(S=\{1,2,3,4,5,6,7,8,9\}\). Determine the cycle representation of \(\sigma\circ \rho\).

Problem 2

Assume that \(\left(a_n\right)_{n=0}^{\infty}\) is the sequence defined as \(a_0=0\), \(a_1=0\), \(a_2=0\), and \(a_n=5\) for \(n\geq 3\). Let \(F(X)\) be the generating function of the sequence \(\left(a_n\right)_{n=0}^{\infty}\). Determine \(F\left(\frac12\right)\).

Problem 3

A cake must contain between \(0\) and \(3\) apples, between \(0\) and \(3\) bananas, and between \(0\) and \(3\) pears. Write down a polynomial in one variable \(x\) such that the coefficient of \(x^n\) is the number of ways to make the cake with \(n\) pieces of fruit.

Problem 4

Determine the number of permutations on the set \(\{1,2,3,4,\dots, 20\}\) that contain one cycle of length \(12\).

Problem 5

Use the method of generating functions to determine the closed formula for the terms of the sequence \(\left(a_n\right)_{n=0}^{\infty}\) that satisfies the recursion: \[a_0=3, \quad a_1=10,\quad\mbox{ and } \quad a_{n+2}=7a_{n+1}-12a_n.\]

Problem 6

Determine the number of \(50\)-digit numbers whose digits do not include \(0\), the digit \(2\) appears an even number of times, and the digit \(3\) appears an odd number of times.

Problem 7

Determine the number of ways to distribute \(50\) red and \(30\) green apples to \(5\) kids if each kid must get at least one green apple and each kid must get at least as many red apples as green apples.