\documentclass[12pt]{article} \usepackage{amsmath} \begin{document} \centerline{\bf Homework 10, MORALLY Due 10:00AM April 29} \newcommand{\SPS}{{\rm SPS}} \newcommand{\IO}{\exists^\infty} \newcommand{\st}{\mathrel{:}} \newcommand{\goes}{\Rightarrow} \newcommand{\NSQ}{{\rm NSQ}} \newcommand{\into}{{\rightarrow}} \newcommand{\lf}{\left\lfloor} \newcommand{\rf}{\right\rfloor} \newcommand{\lc}{\left\lceil} \newcommand{\rc}{\right\rceil} \newcommand{\Ceil}[1]{\left\lceil {#1}\right\rceil} \newcommand{\ceil}[1]{\left\lceil {#1}\right\rceil} \newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor} \newcommand{\Z}{{\sf Z}} \newcommand{\N}{{\sf N}} \newcommand{\Q}{{\sf Q}} \newcommand{\R}{{\sf R}} \newcommand{\D}{{\sf D}} \newcommand{\Rpos}{{\sf R}^+} \begin{enumerate} \item (25 points) Bill makes his Gnilrad lunch that consists of the following: \begin{itemize} \item Sandwich: Egg Salad OR Tuna Fish OR Cheese. \item Fruit: Apple OR Orange OR Grapes OR Blueberries OR Strawberries. \item Desert: Apple Sauce OR Cookie. \end{itemize} For the questions below show your work but also give us an actual number like 10 and not just the notation like $\binom{5}{3}$. \begin{enumerate} \item (5 points) How many ways can Bill make lunch for his Gnilrad? NOTE: She has ONE sandwitch, ONE fruit and ONE desert. \item (10 points) One day she complains: {\it I don't want to have my Fruit be an apple, and my desert be Applesauce at the same time, though I am okay with having one or the other.} Bill obeys her wishes. NOW how many ways can Bill make lunch for her? NOTE: Examples: Darling is happy with Eggsalad-Apple-Cookie but NOT with Eggssalad-Apple-Applesauce.) \item (10 points) One day she complains: {\it (1) I don't want to have my an apple and applesauce at the same time, AND (2) I want 2 different Sandwiches AND (3) I want 3 different Fruits AND (4) I still just want one Desert.} Bill obeys her wishes. NOW how many ways can Bill make lunch for her? \item (0 points but you must answer it) Why does Bill call her {\it Gnilrad}? \end{enumerate} \newpage (25 points) On the last slide of the lecture {\it The Law of Inclusion and Exclusion} is the law for $A_1,A_2,A_3,A_4$. Its complicated! \begin{enumerate} \item (10 points) What if the following hold \begin{itemize} \item Each $A_i$ has $x_1$ elements. \item Each intersection of TWO sets has $x_2$ elements. \item Each intersection of THREE sets has $x_3$ elements. \item Each intersection of FOUR sets has $x_4$ elements. \end{itemize} Give an expression for $|A_1 \cup A_2 \cup A_3 \cup A_4|$ in terms of $x_1,x_2,x_3,x_4$. It should be much simpler then the general law. \item (15 points) Let $A_1,\ldots,A_n$ be sets. Assume that, for $1\le i\le n$, the intersection of $i$ of these sets has size $x_i$. Give an expression for $|A_1 \cup \cdots \cup A_n|$ in terms of $x_1,\ldots,x_n$. You CANNOT use DOT DOT DOT. You can and should use a summation sign. \end{enumerate} \newpage \newpage \item (25 points) How many solutions are there to the equation $$x_1+x_2+x_3+x_4=100$$ with $x_1\ge 1$, $x_2\ge 2$, $x_3\ge 3$, and $x_4\ge 4$. \newpage \item (25 points) Read or re-read the slides on the Horse Numbers. The numbers $H(n)$ will come up in this problem. For $n\ge 2$. Let $I(n)$ be the number of ways that $n$ horses, $x_1,\ldots,x_n$, can finish a race (so orderings with equalities allowed) that have $x_1