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\begin{center}\textbf{Problem Set 10 \hspace{.7in} Foundations \hspace{.7in} Due 4/25/18} \end{center}

\noindent Four of these problems will be graded, with each problem worth 5 points.  Clear and complete justification is required for full credit.  You are welcome to discuss these problems with anyone and everyone, but must write up your own final submission without reference to any sources other than the textbook and instructor. 

\begin{enumerate}
\item Do the Combinatorics assignment on WeBWorK, available via\\ http://crlinwebwork2.coe.edu/webwork2/MTH-215/Combinatorics .

\item The Banana Theorem: Let A be a set with $n$ elements of $k$ different types (such that elements of the same type are regarded as indistinguishable from one another for purposes of orderings).  Let $n_i$ be the number of elements of type $i$ for each integer $i$ from 1 to $k$.  Then the number of different arrangements of the elements in A will be $\frac {n!} {\prod_{i=1}^k \left( n_i!\right)}$.

\item How many distinguishable ways can the letters in the word onamonapia be arranged?

\item If a jar contains six balls, two red, three blue, and one white, and two balls are drawn at random from the jar (without replacement), what is the probability that neither ball is blue?

\item If a jar contains six balls, two red, three blue, and one white, and two balls are drawn at random from the jar (without replacement), what is the probability that neither ball is blue if you know that the white ball is one of the two drawn?

\item $\forall y \in N, 0 \cdot y = y \cdot 0.$

\item $\forall x, y \in N, x \cdot y = y \cdot x \Rightarrow x' \cdot y = y \cdot x'.$

\item $\forall x, y \in N, x \cdot y = y \cdot x.$

\item Using the definition of $S(A)$ from section 5.6, write $S(\varnothing), S(S(\varnothing)), S(S(S(\varnothing)))$, and $S(S(S(S(\varnothing))))$ explicitly.  How many elements are in each of these sets?

\item With the understanding that $0' = 1, 1' = 2, 2' = 3$, and $3' = 4$, where these are elements of a Peano system, show that $2 + 2 = 4$.

\end{enumerate}

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