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\begin{document}

$\;$\hfill Due: 2011/09/28 before class

\begin{center}
{\LARGE\bf Homework 2}
\end{center}

\begin{hw}
We randomly generate numbers $a_1, a_2, ..., a_n$ such that each $a_i$ is uniformly and independently
sampled from $[100]$. What is the probabilty that $\prod_1^n a_i$ is a multiple of 10?
\end{hw}

\begin{hw}
Let $A$ be the $6 \times 6$ matrix where the $i$-th row and $j$-th column is $1$ if $i | j$, otherwise $0$. (Here the row and column indices start from 1.)

Write down $A$ explicitly, and compute $A^{-1}$.
\end{hw}

\begin{hw}
Let $n$ be a positive integer, determine $\sum_{1 \leq m \leq n} \mu(m) \lfloor n / m \rfloor$. Prove your answer.
\end{hw}

\begin{hw}
Count the number of permutations $x_1, x_2, \cdots, x_{2n}$ of $[2n]$ such that $x_i + x_{i+1} \neq 2n+1$ for all $1 \leq i \leq 2n-1$.
\end{hw}

\end{document}

