What is the number of arrangements of all the seven letters of the word $EXAMPLE$ in which the vowels are all separated?
I know that $2520$ is the number of arrangements if there are no restrictions. But what I am asking is how many arrangements are there when no two vowels are next to each other. Thanks.
We first place the letters that are not vowel as, $$\_X\_M\_P\_L\_$$ Now, we have two $E$'s and one $A$ left and five places to place them. But since $E$'s are identical, we have $\frac{5\cdot4\cdot3}{2} = 30$ ways to place them. We also have $4! = 24$ ways to rearrange the consonant letters so the answer should be $30\cdot24=720$.