  Enumeration problems related to words avoiding patterns as well as
permutations that contain the pattern $123$ exactly once have been studied in
great detail. However, the problem of enumerating words that contain the
pattern $123$ exactly once is new and will be the focus of this paper.
Previously, Doron Zeilberger provided a shortened version of Alexander
Burstein's combinatorial proof of John Noonan's theorem that the number of
permutations with exactly one $321$ pattern is equal to $\frac{3}{n}
\binom{2n}{n+3}$. Surprisingly, a similar method can be directly adapted to
words. We are able to use this method to find a formula enumerating the words
with exactly one $123$ pattern. Further inspired by Nathaniel Shar and
Zeilberger's paper on generating functions enumerating 123-avoiding words with
$r$ occurrences of each letter, we examine the algebraic equations for
generating functions for words with $r$ occurrences of each letter and with
exactly one $123$ pattern.
