Fractions and Binomials

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The binomial coefficient, \(\binom{n}{k}\), is defined by the expression:
\[
    \binom{n}{k} = \frac{n!}{k!(n-k)!}
\]
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\usepackage{amsmath}

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% Using the geometry package to reduce
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\usepackage[textwidth=8cm]{geometry}
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Fractions can be used inline within the paragraph text, for 
example \(\frac{1}{2}\), or displayed on their own line, 
such as this:
\[\frac{1}{2}\]
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\documentclass{article}
% Using the geometry package to reduce
% the width of help article graphics
\usepackage[textwidth=8cm]{geometry}
\usepackage{amsmath}% For the \text{...} command
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We use the \texttt{amsmath} package command
\verb|\text{...}| to create text-only fractions
like this:

\[\frac{\text{numerator}}{\text{denominator}}\]

Without the \verb|\text{...}| command the result 
looks like this:

\[\frac{numerator}{denominator}\]
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\documentclass{article}
% Using the geometry package to reduce
% the width of help article graphics
\usepackage[textwidth=9.5cm]{geometry}
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Fractions typeset within a paragraph typically look like this: \(\frac{3x}{2}\). You can force \LaTeX{} to use the larger display style, such as \( \displaystyle \frac{3x}{2} \), which also has an effect on line spacing. The size of maths in a paragraph can also be reduced: \(\scriptstyle \frac{3x}{2}\) or \(\scriptscriptstyle \frac{3x}{2}\). For the \verb|\scriptscriptstyle| example note the reduction in spacing: characters are moved closer to the \textit{vinculum} (the line separating numerator and denominator).

Equally, you can change the style of mathematics normally typeset in display style:

\[f(x)=\frac{P(x)}{Q(x)}\quad \textrm{and}\quad \textstyle f(x)=\frac{P(x)}{Q(x)}\quad \textrm{and}\quad \scriptstyle f(x)=\frac{P(x)}{Q(x)}\]
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\documentclass{article}
% Using the geometry package to reduce
% the width of help article graphics
\usepackage[textwidth=9.5cm]{geometry}
% Load amsmath to access the \cfrac{...}{...} command
\usepackage{amsmath}
\begin{document}
Fractions can be nested but, in this example, note how the default math styles, as used in the denominator, don't produce ideal results...

\[ \frac{1+\frac{a}{b}}{1+\frac{1}{1+\frac{1}{a}}} \]

\noindent ...so we use \verb|\displaystyle| to improve typesetting:

\[ \frac{1+\frac{a}{b}} {\displaystyle 1+\frac{1}{1+\frac{1}{a}}} \]

Here is an example which uses the \texttt{amsmath} \verb|\cfrac| command:

\[
  a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cdots}}}
\]

Here is another example, derived from the \texttt{amsmath} documentation, which demonstrates left
and right placement of the numerator using \verb|\cfrac[l]| and \verb|\cfrac[r]| respectively:
\[
\cfrac[l]{1}{\sqrt{2}+
\cfrac[r]{1}{\sqrt{2}+
\cfrac{1}{\sqrt{2}+\dotsb}}}
\]
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\documentclass{article}
\usepackage{amsmath}
\begin{document}
\newcommand*{\contfrac}[2]{%
{
  \rlap{$\dfrac{1}{\phantom{#1}}$}%
  \genfrac{}{}{0pt}{0}{}{#1+#2}%
}
}
\[
  a_0 +
  \contfrac{a_1}{
  \contfrac{a_2}{
  \contfrac{a_3}{
  \genfrac{}{}{0pt}{0}{}{\ddots}
  }}}
\]

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