Question

5) Quantos anagramas podem ser formados com a palavra MARAJOARA,





a) Começando por M e terminando por A.

b) começando por M

c) terminado por A.

​

Answer 1

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Permutação com elementos repetidos

$\huge\boxed{\boxed{\boxed{\boxed{\sf P_n^{\alpha,\beta,\theta...}=\dfrac{n!}{\alpha!\cdot\beta!\cdot\theta!\cdot...}}}}}$

$\tt a)\\\sf\underbrace{\sf\boxed{\sf M}ARAJOAR\boxed{\sf A}}_{\sf P_7^{3,2}}=\dfrac{7!}{3!\cdot 2!}\\\sf P_7^{3,2}=\dfrac{7\cdot6\cdot5\cdot\diagup\!\!\!4^2\cdot\diagup\!\!\!\!\!\!3!}{\diagup\!\!\!\!\!\!\!3!\cdot\diagup\!\!\!2}\\\sf P_7^{3,2}=7\cdot6\cdot5\cdot2\\\huge\boxed{\boxed{\boxed{\boxed{\sf P_7^{3,2}=420~anagramas}}}}\checkmark$

$\tt b)\\\sf\underbrace{\sf\boxed{\sf M}ARAJOARA}_{\sf P_8^{4,2}}=\dfrac{8!}{4!\cdot2!}\\\sf P_8^{4,2}=\dfrac{8\cdot7\cdot\diagup\!\!\!6^3\cdot5\cdot\diagup\!\!\!\!\!\!4!}{\diagup\!\!\!\!\!\!4!\cdot\diagup\!\!\!2}\\\sf P_8^{4,2}=8\cdot7\cdot6\cdot3\cdot5\\\huge\boxed{\boxed{\boxed{\boxed{\sf P_8^{4,2}=5040~anagramas}}}}\checkmark$

$\tt c)\\\sf\underbrace{\sf MARAJOAR\boxed{\sf A}}_{\sf P_8^{3,2}}=\dfrac{8!}{3!\cdot2!}\\\sf P_8^{3,2}=\dfrac{8\cdot7\cdot6\cdot5\cdot\diagup\!\!\!4^2\cdot\diagup\!\!\!\!\!\!3!}{\diagup\!\!\!\!\!\!3!\cdot\diagup\!\!\!2}\\\sf P_8^{3,2}=8\cdot7\cdot6\cdot5\cdot2\\\huge\boxed{\boxed{\boxed{\boxed{\sf P_8^{3,2}=3360~anagramas}}}}\checkmark$