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\begin{document}

\title{Binomial Coefficients}


\author{Rikki}
\affil{Affiliation not available}



\date{\today}

\maketitle




Binomial coefficients in discrete mathematics are
denoted~\({n \choose k}\), where we
say~\(n\)~choose~\(k.\)~The
variable,~\(n,\)~is usually known as the upper index while
the variable,~\(k,\)~is known as the lower index. Binomial
coefficients are the same amount of combinations
of~\(k\)~items that can be chosen from a set
of~\(n\)~items where order does not matter.
Therefore,~\({n \choose k}\)~gives the~\(k\)~subsets that
are possible out of~\(n\)~total items. When looking further
into binomial coefficients, we notice that the values are
non-negative.~\\\selectlanguage{english}
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The reason why these numbers are called binomial coefficients is because
the numbers themselves appear as coefficients in the expansion of a
binomial,~\(\left(1+d\right)^n,\)~in growing powers of \(d.\)~\\

There are also a few good things to take note of when looking at an
expansion:\\

\begin{itemize}
\tightlist
\item
  There are~\(n+1\)~terms in the
  expansion~\(\left(x+y\right)^n.\)\\
\item
  The degree of each term is~\(n.\)\\
\item
  The powers on~\(x\)~begin with~\(n\)~and
  decrease to~\(0.\)\\
\item
  The powers on~\(y\)~begin with~\(0\)~and
  increase to~\(n.\)\\
\item
  The coefficients are symmetric.~\\
\end{itemize}

In addition, there are also numerous identities that are satisfied by
binomial coefficients such as:\\\selectlanguage{english}
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\caption{{The binomial coefficient can also be described as a coefficient of a
monomial such as,~\(a^x\)~in an expansion
of~\(\left(1+a\right)^n.\)~The exact coefficient in fact appears in the
binomial formula as well only if~\(k\le n.\)%
}}
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\caption{{Notice, this is valid for any elements \(a\)~and
\(b\)~in a commutative ring. Thus, this explains the name
for binomial coefficient. ~ ~ ~ ~%
}}
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\textbf{Suggested Exam Problem:}~A code consists of two digits from 0-5
along with two letters from the alphabet. If repetition of numbers are
allowed but repetition of the letters are not allowed, how many
different codes can be created?\\

References:\\

\url{http://www.mathwords.com/b/binomial\_coefficients.htm}\\

\url{https://www.wu.ac.at/fileadmin/wu/d/i/statmath/Dateien/BinomialCoefficient.pdf}\\

\url{http://mathworld.wolfram.com/BinomialCoefficient.html}\\

\url{http://www.purplemath.com/modules/binomial.htm}

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