Question

Quantos são os anagramas da palavra PIRACICABA que começam com A.

a) 10!
b) 9!/3!2!2!
c) 45360
d) 3024
e) 362.000

Answer 1

$\large\boxed{\begin{array}{l}\sf A\,palavra\,PIRACICABA\,tem\,10\,letras\\\sf sendo\,que\,a\,letra\,A\,se\,repete\,3\,vezes\\\sf a\,letra\,I\,duas\,vezes\\\sf e\,a\,letra\,C\,duas\,vezes.\\\sf iniciando\,por\,A\,temos\,9\,letras\,para\,permutar\\\sf sendo\,que\,as\,letras\,A,\,C\,e\,I\,se\,repetem\,duas\,vezes.\\\sf portanto\\\sf \end{array}}$

$\large\boxed{\begin{array}{l}\sf P_9^{2,2,2}=\dfrac{9!}{2!\cdot2!\cdot2!}=\dfrac{9\cdot8\cdot7\cdot6\cdot5\cdot\backslash\!\!\!4\cdot3\cdot\diagdown\!\!\!\!\!2!}{\diagdown\!\!\!\!\!2!\cdot\backslash\!\!\!2\cdot\backslash\!\!\!2}\\\\\sf P_9^{2,2,2}=45360\end{array}}$