What is the number of arrangements of all the seven letters of the word “EXAMPLE” in which the vowels are all separated? 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 $30\cdot24=720$.
