Question

Quantas são as permutações da palavra metrópoles ?

Answer 1

$\large\boxed{\begin{array}{l}\rm A\,palavra\,METR\acute OPOLES\,possui\,10\,letras\\\rm onde\,se\,repetem\,duas\,letras\,E\,e\,duas\,letras\,O.\\\rm trata-se\,de\,uma\,permutac_{\!\!,}\tilde ao\,com\,repetic_{\!\!,}\tilde ao \\\rm P_n^{\alpha,\beta,\gamma\,\dotsc}=\dfrac{n!}{\alpha!\cdot\beta!\cdot\gamma!\dotsc}\end{array}}$

$\large\boxed{\begin{array}{l}\underbrace{\rm METR\acute OPOLES}_{\rm P_{10}^{2,2}}\\\\\rm P_{10}^{2,2}=\dfrac{10!}{2!\cdot2!}\\\\\rm P_{10}^{2,2}=\dfrac{10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot\backslash\!\!\!4^2\cdot3\cdot\backslash\!\!\!2\!!}{\backslash\!\!\!\!2\!!\cdot\backslash\!\!\!2}\\\\\rm P_{10}^{2,2}=10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot2\cdot3\\\\\rm P_{10}^{2,2}=907200\end{array}}$