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Problem 679

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FREEFAREA

Let $S$ be the set consisting of the four letters
${\texttt{A'},\texttt{E’},\texttt{F'},\texttt{R’}}$.For$n\ge 0$,let$S^*(n)$denotethesetofwordsoflength$n$consistingoflettersbelongingto$S$.Wedesignatethewords$\texttt{FREE}, \texttt{FARE}, \texttt{AREA}, \texttt{REEF}$ as keywords.

Let $f(n)$ be the number of words in $S^{\ast }(n)$ 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: $\mathtt{\text{FREEFAREA}}$
So, $f(9)=1$.

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

Find $f(30)$.

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FREEFAREA

记$S$为包含四个字母的集合${\texttt{A'},\texttt{E’},\texttt{F'},\texttt{R’}}$。\mathrm{对}\mathrm{于}$n\ge 0$，\mathrm{记}$S^*(n)$\mathrm{为}\mathrm{仅}\mathrm{包}\mathrm{含}$S$\mathrm{中}\mathrm{的}\mathrm{字}\mathrm{母}\mathrm{且}\mathrm{长}\mathrm{度}\mathrm{为}$n$\mathrm{的}\mathrm{字}\mathrm{符}\mathrm{串}\mathrm{所}\mathrm{组}\mathrm{成}\mathrm{的}\mathrm{集}\mathrm{合}。\mathrm{我}\mathrm{们}\mathrm{选}\mathrm{择}$\texttt{FREE}, \texttt{FARE}, \texttt{AREA}, \texttt{REEF}$作为关键词。

记$f(n)$为$S^{\ast }(n)$中恰好包含四个关键词各一次的字符串数目。

在$n=9$时首次出现满足要求的字符串，且唯一的拥有$9$个字母的此类字符串是$\mathtt{\text{FREEFAREA}}$。
因此$f(9)=1$。

已知$f(15)=72863$。

求$f(30)$。

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