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\title{Rule of Product}
\author{Austin Warby}
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The Rule of Product goes by other names such as the Multiplication Principle or Sequential Rule. It is a multiplication based rule one follows in certain counting procedures.
"The rule of product states that if an action can be performed by making $A$ choices followed by $B$ choices, then it can be performed $AB$ ways" (Benjamin, 2009).
In other words suppose there exists a set of $A$ choices and a set $B$ choices, multiplying $AB$ will be the number of ways to do $A$ and $B$. As a side note, notice the word "\textbf{and}", this can often be a queue word for situations were one might find need for The Rule of Product.
Independence of a sets is also required when The Rule of Product is in play. That simply means that a set $A$ and set $B$ from earlier are completely separate.
Moreover, this rule extends to any number of sets. Meaning if we have a set $A$, $B$, $C$, and $D$ they can still be multiplied $ABCD$. \\
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So lets look at a simple example.
Say there is a 3 dial combination lock where each dial is completely separate and each contain the numbers 0 through 9. How many possible combinations can be made between the first, second, and third dials?\selectlanguage{english}
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Before moving forward we should first identify our sets. We will call the first dial's 0 through 9 digits set $A$. We will call the second dial's 0 through 9 digits set $B$. We will call the third dial's 0 through 9 digits set $C$. Are the three sets independent? Yes they are. Note that 0 through 9 represents 10 values or choices for each dial.
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Multiplying these sets together we would have $ABC$ which would be $10\cdot10\cdot10$ or $10^3$ choices. So altogether are 1000 different possible combinations
\textbf{Possible Exam Question}\\
How many license plates can be made if the first 3 entries must be letters, followed by 3 numbers?
\textbf{References}\\
https://en.wikipedia.org/wiki/Rule_of_product \\
http://www.cut-the-knot.org/arithmetic/combinatorics/BasicRules.shtml\\
http://www.cs.cornell.edu/courses/cs280/2004fa/280wk6_x4.pdf\\
https://www.youtube.com/watch?v=sEul6TMYDY0\\
Benjamin, Arthur T. *Discrete Mathematics*. Chantilly: Great Courses, 2009. Print.
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