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<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>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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*Sign up*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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