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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.