\documentclass[a4paper]{article}
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\title{Reverse combinations}
\author{David Wales}
\date{\today}
\begin{document}
\section*{The problem}
There is a randomly generated three-panel comic, with 74000 possible combinations.\footnote{\url{http://pandyland.net/random/}} How many images are there to randomly select from?
\section*{The Solution}
It is easy to go the other way --- If you have $n$ images, and you want to find out how many combinations of $3$ panels there are, you can simply use the following formula:
\begin{equation}
\dbinom{n}{r} = \cfrac{n!}{r!\left( n-r \right)!}.
\end{equation}
So, if we have r = 3, and
\begin{equation}
\dbinom{n}{3} = 74000,
\end{equation}
then
\begin{align}
\cfrac{n!}{3! \left( n-3 \right)!} &= 74000 \\
\cfrac{n(n-1)(n-2) \cancel{(n-3)!}}{3! \cancel{ \left( n-3 \right)!}} &= 74000 \\
n(n-1)(n-2) &= 444000\\
n^3-3n^2+2n - 444000 = 0.
\end{align}
Solving for $n$ using a computer\footnote{\url{http://www.sympygamma.com/input/?i=solve\%28n**3+-+3*n**2+\%2B+2*n+-+444000\%2C+n\%29}} gives
\begin{equation}
n = 77.2932056177889.
\end{equation}
So, there are probably around 77 panels available for the random comic.
\end{document}
