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\headers{Math378}{Homework 12}{Due Mon. Apr 21, 2025}

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\noindent
\textbf{Directions:} Solve the following problems.  All written work must be your own.  See the course syllabus for detailed rules.


\begin{enumerate}
\item {} [5.2.4] Find a simple expression for $\sum_{k=0}^n (2k+1)\binom{n}{k}$.

\item {} Let $n\ge 1$, let $O_n$ be the set of odd numbers in $\{0,1,\ldots,n\}$, and let $E_n$ be the set of even numbers in $\{0,1,\ldots,n\}$.  Give a combinatorial proof that $\sum_{k\in E_n} \binom{n}{k} = \sum_{k\in O_n} \binom{n}{k}$.

\item {} [5.2.9] Let $n\ge 1$, let $O_n$ be the set of odd numbers in $\{0,1,\ldots,n\}$, and let $E_n$ be the set of even numbers in $\{0,1,\ldots,n\}$.  Let $a_n = \sum_{k\in E_n} k\binom{n}{k}$ and $b_n = \sum_{k\in O_n} k\binom{n}{k}$.
\begin{enumerate}
	\item Use the binomial theorem to find expressions for $a_n + b_n$ and $a_n - b_n$.
	\item Find formulas for $a_n$ and $b_n$.
\end{enumerate}

\item {} [8.1.3] How many permutations of the letters in SCRIPPS have no two consecutive letters the same?

\item How many permutations of the letters in AABBCC\ldots ZZ have no two consecutive letters the same?  Find a summation formula.

\item {} [10.1.2] Is $7,7,6,5,4,4,4,3,2$ a graphic sequence?

\end{enumerate}
\end{document}
