I have a statement that says:
With the letters of the word LIBRO, how many words can I form (with or without meaning), that start with a vowel?
My development was:
I have two cases, start with $o$, and start with $i$.
When start with $o$ have $P_4$ differentes words, and same when start with $i$, so applying multiplicative principle, i have $P_4 * P_4 = 576$, but the correct result should be $48$, what is wrong in my development ?
On
To keep things concrete maybe it would help to imagine actually writing down all the possibilities. Start by writing down on paper the possibilities that begin with the letter O. There are 24 such possibilities, as you mentioned, so just write them down.
Next write down the possibilities that start with the letter I. There are 24 such possibilities, as you mentioned, so you will write down 24 words here as well.
And then you're done. So how many words did you write down? 48 words have been written on the page.
It should be $P_4+P_4= 2P_4=2(24)=48$.
Or more explicitly, there are $2$ options for the first letter, $4$ options for the second after the second letter, $3$ options for the third, $2$ options for the fourth, and $1$ option for the last. $2\cdot (4!)=48$.