Question

What is the number of ways one can arrange the letters in the word $GARDEN$ with the vowels in alphabetical order?

Answer 1

There are only two vowels in the word, $A$ and $E$. They should be in alphabetical order.

First place $A$ at the first place, $E$ can occupy any of the remaining $5$ places. Total arrangements:$=5\times 4!$

Next place $A$ in the second place, $E$ can occupy any of the remaining $4$ places. Total arrangements $=4\times 4!$

By repeating this process until $A$ occupies the last but one place. $A$ cannot occupy the last place.

$\therefore$, the number of total arrangements is $4!\times (5+4+3+2+1)=4!\times 15=360$ways.