7.6 - Counting Principles

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<?xml version="1.0" encoding="iso-8859-1"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title>7.6 - Counting Principles</title> <meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" /> <link href=". ./m116.css" rel="stylesheet" type="text/css" /> </head> <body> <h1>7.6 - Counting Principles</h1> <p>Each branch of mathematics has its own fundamental theorem(s). If you check out fundamental in the dictionary, you will see that it relates to the foundation or the base or is elementary. Fundamental theorems are important foundations for the rest of the material to follow.</p> <p>Here are some of the fundamental theorems or principles that occur in your text. </p> <dl> <dt><strong>Fundamental Theorem of Arithmetic</strong> (pg 8)</dt> <dd>Every integer greater than one is either prime or can be expressed as an unique product of prime numbers.</dd> <dt><strong>Fundamental Theorem of Algebra</strong> (pg 264)</dt> <dd>Every polynomial in one variable of degree n&gt;0 has at least one real or complex zero.</dd> <dt><strong>Fundamental Theorem of Linear Programming</strong> (pg 411)</dt> <dd>If there is a solution to a linear programming problem, then it will occur at a corner point, or on a line segment between two corner points.</dd> </dl> <h2>Fundamental Counting Principle</h2> <p>If there are m ways to do one thing, and n ways to do another, then there are m*n ways of doing both. </p> <p>The Fundamental Counting Principle is the guiding rule for finding the number of ways to accomplish two tasks. </p> <h3><strong>Examples using the counting principle:</strong> </h3> <p>Let's say that you want to flip a coin and roll a die. There are 2 ways that you can flip a coin and 6 ways that you can roll a die. There are then 2x6=12 ways that you can flip a coin <em>and</em> roll a die.</p> <p>If you want to hit one note on a piano and play one string on a banjo, then there are 88 * 5 = 440 ways to do both.</p> <p>If you want to draw 2 cards from a standard deck of 52 cards without replacing them, then there are 52 ways to draw the first and 51 ways to draw the second, so there are a total of 52*51 = 2652 ways to draw the two cards.</p> <h2><strong>Sample Spaces</strong></h2> <p>A listing of all the possible outcomes is called the sample space and is denoted by the capital letter S. </p> <p>The sample space for the experiments of flipping a coin and rolling a die are S = { H1, H2, H3,
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H4, H5, H6, T1, T2, T3, T4, T5, T6}. Sure enough, there are twelve possible ways. The fundamental counting principle allows us to figure out that there are twelve ways without having to list them all out.</p> <h2>Permutations</h2> <p>A permutation is an arrangement of objects, without repetition, and order being important. Another definition of permutation is the number of such arrangements that are possible.
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7.6 - Counting Principles - &lt;?xml...

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