Question

Quantos anagramas tem a palavra aritmética.

Answer 1

Aritmetica tem 10 letras, dessas letras:

2 "a's"
2 "i's"
1 "r"
2 "t's"
1 "m"
1 "e"
1 "c"

Entao:

$\dfrac{10!}{2!2!2!1!1!1!1!}=\dfrac{10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2}{2\cdot2\cdot2}=453600$

Answer 2

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Permutação com elementos repetidos

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

$\sf\underbrace{\sf ARITM\acute ETICA}_{\sf P_{10}^{2,2,2}}=\dfrac{10!}{2!\cdot2!\cdot2!}\\\sf P_{10}^{2,2,2}=\dfrac{10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot\diagup\!\!\!\!\!\!2!}{\diagup\!\!\!\!\!\!2!\cdot2\cdot2}\\\sf P_{10}^{2,2,2}=\dfrac{10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot\diagup\!\!\!\!4\cdot3}{\diagup\!\!\!\!4}\\\sf P_{10}^{2,2,2}=10\cdot9\cdot8\cdot7\cdot6\cdot5\cdot3\\\huge\boxed{\boxed{\boxed{\boxed{\sf P_{10}^{2,2,2}=453600~anagramas}}}}\checkmark$