Question

(FCC -BA) Considerem-se todos os anagramas da palavra LÓGICA. Quantos deles tem as vogais juntas ?

Answer 1

$\text{Poss}\mathrm{\acute{i}veis\ arranjos\ das\ vogais:}\\\\\underbrace{\underline{\ \ \ }\ \underline{\ \ \ }\ \underline{\ \ \ }}_{3!}\ \underline{\ \ \ }\ \underline{\ \ \ }\ \underline{\ \ \ }\ \Bigg\vert\ \underline{\ \ \ }\ \underbrace{\underline{\ \ \ }\ \underline{\ \ \ }\ \underline{\ \ \ }}_{3!}\ \underline{\ \ \ }\ \underline{\ \ \ }\ \Bigg\vert\ \underline{\ \ \ }\ \underline{\ \ \ }\ \underbrace{\underline{\ \ \ }\ \underline{\ \ \ }\ \underline{\ \ \ }}_{3!}\ \underline{\ \ \ }\ \Bigg\vert\ \underline{\ \ \ }\ \underline{\ \ \ }\ \underline{\ \ \ }\ \underbrace{\underline{\ \ \ }\ \underline{\ \ \ }\ \underline{\ \ \ }}_{3!}\\\\3!+3!+3!+3!=6+6+6+6=24\\\\\\\\\text{Os poss}\mathrm{\acute{i}veis\ arranjos\ das\ consoantes\ s\tilde{a}o\ as\ permuta\text{\c{c}}\tilde{o}es}\\\mathrm{\ dos\ espa\text{\c{c}}os\ restantes\ dos\ arranjos\ das\ vogais.}\\\mathrm{\ Assim\ sendo,\ temos\ a\ seguinte\ quantidade\ de\ arranjos\ das\ consoantes:}\\\\3!+3!+3!+3!=24\\\\\\\text{A quantidade de anagramas com vogais juntas}\mathrm{\ \acute{e}\ dado\ pela}\\\mathrm{\ multiplica\text{\c{c}}\tilde{a}o\ das\ quantidades\ dos\ arranjos\ das\ vogais\ e\ consoantes:}\\\\24\times24=\boxed{\boxed{576}}$