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Freefarea

Published on Sunday, 15th September 2019, 01:00 am; Solved by 350

Problem 679

Let $S$ be the set consisting of the four letters $\{\texttt{`A'},\texttt{`E'},\texttt{`F'},\texttt{`R'}\}$.
For $n\ge 0$, let $S^*(n)$ denote the set of words of length $n$ consisting of letters belonging to $S$.
We designate the words $\texttt{FREE}, \texttt{FARE}, \texttt{AREA}, \texttt{REEF}$ as keywords.

Let $f(n)$ be the number of words in $S^*(n)$ that contains all four keywords exactly once.

This first happens for $n=9$, and indeed there is a unique 9 lettered word that contain each of the keywords once: $\texttt{FREEFAREA}$
So, $f(9)=1$.

You are also given that $f(15)=72863$.

Find $f(30)$.