Question

Considerando todos os anagramas da palavra
PATATIVA. Quantos anagramas começam por vogal?

a) 3360

b) 1260

c) 420

d) 1680

e) 450​

Answer 1

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$\sf comec_{\!\!,}ando~pela~letra~A:\\\sf\underbrace{\boxed{A}PT\!AT\!IV\!A}_{P_7^{2,2}}=\dfrac{7!}{2!\cdot2!}\\\sf P_7^{2,2}=\dfrac{7\cdot6\cdot5\cdot\diagup\!\!\!4^2\cdot3\cdot\diagup\!\!\!\!2!}{\diagup\!\!\!\!2!\cdot\diagup\!\!\!2}\\\sf P_7^{2,2}=7\cdot6\cdot5\cdot2\cdot3=1260.$

$\sf comec_{\!\!,}ando~pela~letra~i:\\\underbrace{\boxed{I}P\!ATT\!AV\!\!A}_{P_7^{2,3}}=\dfrac{7!}{2!\cdot3!}\\\sf P_7^{2,3}=\dfrac{7\cdot6\cdot5\cdot\diagup\!\!\!\!4^2\cdot\diagup\!\!\!\!3!}{\diagup\!\!\!2\cdot\diagup\!\!\!\!3!}\\\sf P_7^{2,3}=7\cdot6\cdot5\cdot2\\\sf P_7^{2,3}=420$

$\sf ao~todo~teremos:\\\huge\boxed{\boxed{\boxed{\boxed{\sf1260+420=1680\checkmark}}}}\\\huge\boxed{\boxed{\boxed{\boxed{\sf\maltese~alternativa~d }}}}$