Question

5. Considere os anagramas formados a partir de TOCANTINS.

a) Quantos são?

b) Quantos começam por vogal?

c) Quantos começam e terminam por consoante?

d) Quantos têm as letras TOCA juntas e nessa ordem?​

ME AJUDA PFVR

Answer 1

$\large\boxed{\begin{array}{l}\sf A\,palavra\,TOCANTINS\,possui\,9\,letras\\\sf com\,repetic_{\!\!,}\tilde ao\,de\,duas\,letras\,T\,e\,duas\,letras\,N.\\\rm a )~\sf\underbrace{\sf TOCANTINS}_{P_9^{2,2}}\\\\\sf P_9^{2,2}=\dfrac{9!}{2!\cdot 2!}=\dfrac{9\cdot8\cdot7\cdot6\cdot5\cdot\backslash\!\!\!4^2\cdot3\cdot\diagdown\!\!\!\!\!\!2!}{\diagdown\!\!\!\!\!\!2!\cdot\backslash\!\!\!2}\\\\\sf P_9^{2,2}=90720\,anagramas\end{array}}$

$\large\boxed{\begin{array}{l}\rm b)~\sf Para\,iniciar\,por\,vogal\,temos\,3\,possibilidades.\\\sf Escolhida\,uma\,vogal\,teremos\,8\,letras\,para\,permutar\\\sf sendo\,que\,2\,letras\,T\,e\,2\,letras\,N\,se\,repetir\tilde ao.\\\sf dessa\,forma\\\sf 3\cdot P_8^{2,2}=3\cdot\dfrac{8!}{2!\cdot2!}=\dfrac{3\cdot8\cdot7\cdot6\cdot5\cdot\diagup\!\!\!\!4^2\cdot3\cdot\diagdown\!\!\!\!\!2!}{\diagdown\!\!\!\!\!2!\cdot\diagup\!\!\!2}\\\\\sf 3\cdot P_8^{2,2}=30240\end{array}}$

$\large\boxed{\begin{array}{l}\rm c)\\\underline{\rm Consoantes\,que\,comec_{\!\!,}am\,por\,T:}\\\sf P_8^{2}=\dfrac{8!}{2!}=\dfrac{8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot\diagdown\!\!\!\!\!2!}{\diagdown\!\!\!\!\!2!}\\\\\sf P_8^2=20160\end{array}}$

$\large\boxed{\begin{array}{l}\underline{\rm Consoantes\,que\,comec_{\!\!,}am\,por\,N\!:}\\\sf P_8^2=\dfrac{8!}{2!}=20160\end{array}}$

$\large\boxed{\begin{array}{l}\underline{\rm Consoantes\,que\,comec_{\!\!,}am\,por\,C\!:}\\\sf P_8^{2,2}=\dfrac{8!}{2!\cdot 2!}=\dfrac{8!}{2!\cdot2}=\dfrac{1}{2}\cdot20160=10080\\\underline{\rm Consoantes\,que\,iniciam\,por\,S\!:}\\\sf P_8^{2,2}=\dfrac{8!}{2!\cdot 2!}=10080.\\\underline{\rm Ao\,todo\,teremos:}\\\sf20160+20160+10080+10080=60480\end{array}}$

$\large\boxed{\begin{array}{l}\sf Quando\,as\,letras\, T,O,C\,e\,A\,ficarem\,juntas\,contar\tilde ao\\\sf como\,uma\,s\acute o\,letra.\\\sf Assim\,teremos\\\sf\underbrace{\sf\boxed{\sf TOCA}NTINS}_{P_6^2}\\\\\sf P_6^2=\dfrac{6!}{2!}=\dfrac{6\cdot5\cdot4\cdot3\cdot\diagdown\!\!\!\!2\!!}{\diagdown\!\!\!\!\!2!}=360\end{array}}$