Question

Quantos são os anagramas da palavra ENGENHARIA?
a) 453.600
b) 3.628.800
c) 1.814.400
d) 604.800
e) 151.200

Answer 1

$\boxed{\begin{array}{l}\sf a~palavra~ENGENHARIA~possui~10~letras\\\sf com~repetic_{\!\!,}\tilde ao~de~duas~letras~E,2~letras~N~e~duas~letras~A\\\sf portanto~trata-se~de~uma~permutac_{\!\!,}\tilde ao~de~10~letras\\\sf com~repetic_{\!\!}\tilde ao~de~duas~letras~E,duas~letras~N~e~duas~letras~A.\end{array}}$

$\Large\boxed{\begin{array}{l}\sf\underbrace{\sf ENGENHARIA}_{\sf P_{10}^{2,2,2}}=\dfrac{10!}{2!\cdot 2!\cdot 2!}\\\sf P_{10}^{2,2,2}=\dfrac{10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot\diagup\!\!\!\!2!}{\diagup\!\!\!\!\!\!2!\cdot2\cdot1\cdot2\cdot1}\\\sf P_{10}^{2,2,2}=\dfrac{10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot\backslash\!\!\!4\cdot3}{\backslash\!\!\!4}\\\sf P_{10}^{2,2,2}=151~200~anagramas\\\huge\boxed{\boxed{\boxed{\boxed{\sf\maltese~alternativa~e}}}}\end{array}}$