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\title{CS200 - Problem Set 7\\\small{Due: Monday, April 9 to Canvas before class}}
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\begin{enumerate}
\item Big-O Proofs
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\item\textbf{[11 points]} Prove that $5-10x+2x^2=O(x^2)$.
\item\textbf{[11 points]} Prove that $\log_3(n^2)=O(\log_2(n)).$ (This is a good question to review properties of logarithms. We will be using logarithms later in this class, and also they are important in many areas of computer science. Hints: recall if $\log_b(c)=x$ this means $b^x=c$. As a consequence, $a=b^{\log_b(a)}$. Also, $\log_b(a\times c)=\log_b(a)+\log_b(c)$. To prove this result, try to change the base of the term $\log_3(n^2)$ from $3$ to $2$. If you are feeling uncomfortable with this problem, go online and find extra practices problems dealing with logarithms and exponentiation.)
\item\textbf{[11 points]} Prove the following statement is false: $2^{2n}=O(2^n).$ (Hint: try a proof by contradiction!)
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\item \textbf{[6 points]} Explain how you could use a graph to represent e-mail messages sent between employees at at company. What should the vertices and edges represent? Should edges be directed or not directed? Should there be self-loops in the graph (edges from one vertex back to itself)? Should there be multiple edges allowed between two vertices?
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\item Consider graphs on the vertices $\{a,b,c,d,e,f\}$ such that each vertex is connected to exactly one other vertex by an edge.
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\item \textbf{[6 points]} How many possible undirected graphs are there that satisfy the above condition? (Hint: first think about how many choices you have for the vertex connected to $a$.)
\item \textbf{[6 points]} If the graph is directed, how does your answer to part (a) change?
\item \textbf{[3 points]} Come up with a real world application of such a graph, where knowing the number of possible graphs you could create would be helpful.
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\item
Using binomial coefficients (and perhaps some other counting rules), determine how many bit strings of length 10 have
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\item \textbf{[6 points]}
exactly three 0s?
\item \textbf{[6 points]}
at least seven 1s?
\item \textbf{[6 points]}
exactly three zeros or start with a 1?
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(Your answer may contain terms of the form $\binom{a}{b}$)
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\item How long did you spend on this homework?
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