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\begin{document}
\title{Weekly Blog Post \#2 - How Many?}
\author[1]{TJ}%
\affil[1]{Affiliation not available}%
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\date{\today}
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\section*{Introduction}
{\label{760146}}\par\null
Every day there are opportunities to try to figure out the possible
number of outcomes for any given series of events. Poker hands, ice
cream cones, lottery ticket order, combination locks, and just general
options in your own closet are examples of how we can use different
forms of counting each day! The basic concept of counting and combining
numbers of things stems from the general phrase, ``How many\ldots{}?''.
The details in the question often determine how we will proceed with
evaluating the problem. Situations of whether or not the numbers are
allowed to repeat and if the order matters or not will help to choose
the correct way to establish the combinations.
\par\null
\section*{The Sum and Product Rules}
{\label{829955}}
\section*{}
{\label{829955}}
The sum and product rules are the fundamental rules of counting. ~
The rule of sum states that if an action is performed by making A choice
or B choice, then it can be performed A + B ways. Here are some examples
of simple rule of sum situations:
\begin{itemize}
\tightlist
\item
If there are 12 boys~and 8 girls in my class, then I have 20 students
to choose to call on.
\item
There are 3 cats, 5 dogs, and 2 birds at the pet shop, I have 10 pets
to choose from.
\end{itemize}
\par\null
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 in AB ways.
Here are some examples of simple rule of product situations:
\begin{itemize}
\tightlist
\item
If I have 5 skirts and 10 shirts, then I can create 50 different
outfits.
\end{itemize}
\par\null
\section*{\texorpdfstring{Binomial Coefficient and
\emph{n!}}{Binomial Coefficient and n!}}
{\label{966618}}\par\null
So what happens when the ``how many'' has specific restrictions? Does
the order of the numbers matter, ~or can the objects be repeated? When
these situations occur, the rule of productis is expanded upon. Concepts
such as \emph{n!~}and the binomial coefficient are used to solve such
scenarios. Here is a table that shows how the numbers will be combined
based on the specifics of the how many question.
\par\null
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
& Order matters & Order does not matter \\ \hline
& sequences & multisubsets \\
Repetition is allowed & $n^k$ & $(\frac{n + k - 1}{k})$ \\
& & \\\hline
& permutations & combinations \\
Repetition is not allowed & $n!/ (n - k)!$ & $\frac{n!}{k! (n - k)!}$ \\
& & \\\hline
\end{tabular}
\end{center}
\par\null
The all-important question, `How many ways are there to select k objects
from n distinct objects?' can now be broken down to either sequences,
permutations (arrangements), combinations (subsets), or multi subsets.
The formulas that each situation requires are worth remembering, once we
understand these basic concepts, we can determine many counting and
probability questions!
\par\null
\section*{Exam Questions}
{\label{285916}}\par\null
1. ~In the ABC school district, student ID numbers consist of 4 numbers
followed by 2 letters. How many possible student ID numbers are there?
\par\null
2. Six friends want to sit in a row together at the assembly. If there
are six spots left in the back row, how many ways can the friends sit
together?
\par\null
\subsection*{References}
{\label{991266}}
Benjamin, A.T. (2009). Discrete Mathematics. Chantilly, VA: The Teaching
Company.
\par\null
Rule of Sum and Rule of Product Problem Solving.
(2018).~\emph{brilliant.org}. Retrieved Jan 30, 2018,
from~\url{https://brilliant.org/wiki/rule-of-sum-and-rule-of-product-problem-solving/}
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