Question

How many ways are there to arrange the letters in the word GARDEN with the vowels in alphabetical order?
(a) 120
(b) 240
(c) 360
(d) 480

Answer 1

Hint: Divide the total ways of arrangement of all the letters by ways of arrangement of vowels.

Complete step-by-step answer:
We are given a word GARDEN. We have to find the number of ways to arrange the letters in this word with the vowels in alphabetical order.
The total number of letters in given word = 6 {G, A, R, D, E, N}
Therefore, the total ways in which all letters can be arranged is
\[\underline{6}\times \underline{5}\times \underline{4}\times \underline{3}\times \underline{2}\times \underline{1}\]
1. There are a total 6 ways to fill the first place.
2. Now, the second place can be filled by remaining 5 letters
3. Third place can be filled by remaining 4 letters
4. Similarly, fourth, fifth and sixth places can be filled by remaining 3, 2 and 1 letters respectively.
Therefore, we get total ways in which all letters can be arranged
\[\begin{align}
  & =6! \\
 & =6\times 5\times 4\times 3\times 2\times 1 \\
 & =720 \\
\end{align}\]
Now, total number of vowels in the given word = 2 (A, E)
The total ways in which these vowels can be arranged \[=\underline{2}\times \underline{1}\]
1. There are 2 ways to fill the first place.
2. Now second place can be filled in just one way.
Therefore, we get the total ways in which these vowels can be arranged = 2! = 2
Now, the total number of ways in which the letters can be arranged such that vowels are in alphabetical order are
\[\begin{align}
  & =\dfrac{\text{Total number of ways of arrangements of all letters}}{\text{Total number of ways of arrangement of vowels}} \\
 & =\dfrac{6!}{2!}=\dfrac{720}{2}=360 \\
\end{align}\]
Hence, option (c) is correct.

Note: Students can also solve this question in this way.
_ _ _ _ _ _
First we select 2 places out of 6 places in which vowels will be arranged in alphabetical order that is only 1 way (A, E)
Therefore, the total number of ways of selecting 2 out of 6 places for vowels in alphabetical order are \[6{{C}_{2}}\].
\[\begin{align}
  & =\dfrac{6!}{2!4!}=\dfrac{6\times 5\times 4!}{2\times 4!} \\
 & =3\times 5=15\text{ ways} \\
\end{align}\]
Now, remaining 4 words (G, R, D, N) will be arranged in 4 remaining places such that number of ways will be
\[=4!=4\times 3\times 2\times 1=24\]
Therefore, the total number of ways of arranging the letters of word such that the vowels are in alphabetical order
\[\begin{align}
  & =6{{C}_{2}}.4! \\
 & =15\times 24 \\
 & =360\text{ ways} \\
\end{align}\]
Hence, option (c) is correct.