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\title{General Mathematics and Computational Science I}
\author{Exercise 9}
\date{October 11, 2007}
\maketitle

\begin{enumerate}

\item (From Ivanov, p.\ 20; note that there is a typo in the book!)

Let $S(n,n-1)=1$ and $(n-1)\,(k-1)\,S(n,k)=(n-k) \, S(n,k-1)$.  Deduce
that
\begin{equation*}
  S(n,k) = \binom{n-2}{k-1} \, (n-1)^{n-k-1} \,.
\end{equation*}


\item (From Ivanov, p.\ 21.) Prove the inequality
\begin{equation*}
  \binom{n}{k} \leq \binom{n}{[n/2]}
\end{equation*}
for $k=0,\dots,n$.

\item In how many ways can you write the number $4$ as the sum of $5$
nonnegative integers?
\end{enumerate}
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