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\title{General Mathematics and Computational Science I}
\author{Exercise 6}
\date{October 12, 2006}
\maketitle

\begin{enumerate}

\item A company has $2n$ employees.
\begin{enumerate}
\item In how many ways can they be divided into $n$ groups of two
members each?
\item In how many ways can they be divided into two departments of $n$
members each, if each department must choose a president and a
vice-president?
\end{enumerate}

\item (From Ivanov, p.\ 17.)
\begin{enumerate}
\item Verify, by explicit computation, that the binomial coefficients
\begin{equation*}
  \binom{n}{k} = \frac{n!}{k! \, (n-k)!}
\end{equation*}
satisfy the recursion relation
\begin{equation*}
  \binom{n+1}{k} = \binom{n}{k} + \binom{n}{k-1} \,. \tag{*}
\end{equation*}
\item Derive this recursion relation from the interpretation of the
binomial coefficients as the ``number of $k$-element subsets of an
$n$-element set''.
\end{enumerate}

\item (From Ivanov, p.\ 19.) Prove that
\begin{equation*}
  \binom{2n}{k} = \sum_{l=0}^k \binom{n}{l} \, \binom{n}{k-l} \,.
\end{equation*}

Hint: Use the function $P_n(x)$ from class, and the fact that
\begin{equation*}
  \bigl( (1+x)^n \bigr)^2 = (1+x)^{2n} \,.
\end{equation*}

\end{enumerate}
\end{document}

\item Using the binomial theorem, expand the following binomials:
$(1+1)^n$,$(1-1)^n$, $(1+i)^n$ and $(1-i)^n$, where $i^2=-1$.

\item (From Ivanov, p.\ 18.) Prove the binomial theorem directly by
noting that the coefficients of the monomials $a^k \, b^{n-k}$ in the
expansion of $(a+b)^n$ satisfy the recursion relation (*).

