In how many ways are the letters of the word DOCUMENT arranged so tha…

The correct answer is Option (2) → 4320

Step 1: Identify vowels and consonants

• Vowels: O, U, E → 3 vowels
• Consonants: D, C, M, N, T → 5 consonants

Step 2: Keep all vowels together

Treat the 3 vowels as one block.

So total units to arrange:

• 1 vowel block + 5 consonants = 6 units

Number of ways to arrange these 6 units:

$6! = 720$

Step 3: Arrange vowels within the block

The 3 vowels can be arranged among themselves in:

$3! = 6 \text{ ways}$

Step 4: Total arrangements

$720 \times 6 = 4320$