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\begin{document}

$\;$\hfill Due: 2012/09/26 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}
Draw the Hasse diagram for the divisibility lattice $([12], |)$.
\end{hw}

\begin{hw} Let $n$ be a positive integer, find
\[ \sum_{i = 1}^n (-1)^{i-1} i \binom{n}{i} .\]
Justify your answer.
\end{hw}

\begin{hw}
Give a combinatorial proof for the following equation. For any positive integers
$a$ and $b$,
\[ \sum_{i=0}^a \binom{a}{i} \binom{b+i}{a} = \sum_{i=0}^a \binom{a}{i}\binom{b}{i} 2^i .\]
\end{hw}

\begin{hw}
Let $\cl = (X, \preccurlyeq)$ be a finite lattice. For any $X_1, X_2 \subseteq X$, define
\[ \mu(X_1, X_2) = \sum_{x_1 \in X_1, x_2 \in X_2} \mu(x_1, x_2). \]
Suppose $A$, $B$, and $C$ is a partition of $X$ such that $A$ is an ideal and
$C$ is a filter. Prove that
\[ \mu(A, C) = \mu(B, B) - 1. \]
\end{hw}

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