Question

How many ways are there to arrange the letters in the word $GARDEN$ with the vowels in alphabetical order ?

Answer 1

The word $GARDEN$ has 6 letters.
Therefore, all the ways of permuting the letters of $GARDEN$ are: $6!$
Now, the questions wants the vowels to appear in alphabetical order, this means that in all the words, $A$ must appear before $E$.
There are $2!$ ways of permuting $A$ and $E$ out of which we just want 1, so the final answer is: $6!\cdot \frac{1}{2!}=\frac{720}{2}=360$