Number of ways in which the letters of word GARDEN can be arranged with vowels in alphabetical order, is

Options

A. 360
B. 240
C. 120
D. 480

Answer

To find the number of ways in which the letters of the word "GARDEN" can be arranged with the vowels in alphabetical order, we can follow these steps:

### Step 1: Identify the letters in the word "GARDEN"
The letters in the word "GARDEN" are G, A, R, D, E, N.

### Step 2: Count the total number of letters
The total number of letters in "GARDEN" is 6.

### Step 3: Calculate the total arrangements of the letters
The total number of arrangements of 6 distinct letters can be calculated using the factorial of 6:
\[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \]

### Step 4: Identify the vowels in the word
The vowels in "GARDEN" are A and E.

### Step 5: Arrange the vowels in alphabetical order
The vowels A and E in alphabetical order are A, E. Since we want the vowels to always appear in this order, we will consider them as fixed in their positions.

### Step 6: Calculate the arrangements considering the fixed vowels
Since the vowels A and E must be in alphabetical order, we can treat them as fixed. Therefore, we need to arrange the 6 letters (G, A, R, D, E, N) while ensuring A comes before E.

To find the number of valid arrangements, we can divide the total arrangements by the number of arrangements of the vowels:
\[ \text{Number of arrangements with vowels in order} = \frac{720}{2} = 360 \]

### Conclusion
Thus, the number of ways in which the letters of the word "GARDEN" can be arranged with the vowels in alphabetical order is 360.

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