MHB Number of ways to arrange letters

[evinda]

Hey again! :)

I am given this exercise:
With how many ways can we arrange $5$ letters $A$, $3$ letters $B$ and $4$ letters $C$?

I thought that it is :

$$\frac{(5+3+4)!}{5! \cdot 3! \cdot 4!}=\frac{12!}{5! \cdot 3! \cdot 4!}$$

Could you tell me if it is right? (Blush)

 

[I like Serena]

Hey again! :)

I am given this exercise:
With how many ways can we arrange $5$ letters $A$, $3$ letters $B$ and $4$ letters $C$?

I thought that it is :

$$\frac{(5+3+4)!}{5! \cdot 3! \cdot 4!}=\frac{12!}{5! \cdot 3! \cdot 4!}$$

Could you tell me if it is right? (Blush)

Yep.
It is right! (Mmm)

 

[evinda]

Yep.
It is right! (Mmm)

Nice,thank you! :)

 

[DavidCampen]

I would have said that the answer is:
\[(\begin{matrix}12\\5\end{matrix})(\begin{matrix}7\\4\end{matrix})=(\begin{matrix}12\\4\end{matrix})(\begin{matrix}8\\5\end{matrix})=
(\begin{matrix}12\\3\end{matrix})(\begin{matrix}9\\4\end{matrix}) etc.=\frac{12!}{3!\cdot4!\cdot5!}\]

I don't understand the purpose of Evinda's instructor or textbook asking for so many answers to be computed only without also supplying methods or proofs.