In how many different ways, can the letters of the word ASSOCIATION b…

The correct answer is Option (4) → 32400

Step 1: Count letters and vowels/consonants

Word: A S S O C I A T I O N → 11 letters

• Vowels: A, O, I, A, I, O → 6 vowels (A twice, I twice, O twice)
• Consonants: S, S, C, T, N → 5 consonants (S twice)

Step 2: Treat vowels as a single entity

• Consider all vowels together as one block.
• So total "letters" to arrange = 5 consonants + 1 vowel block = 6 items
• Consonants have S repeated twice.

Arrangements of 6 items:

$\frac{6!}{2!} = \frac{720}{2} = 360$

Step 3: Arrange vowels inside the block

• Vowels: A, O, I, A, I, O → 6 letters with A, I, O repeated twice each

$\text{Arrangements} = \frac{6!}{2! \cdot 2! \cdot 2!} = \frac{720}{8} = 90$

Step 4: Total arrangements

$360 \times 90 = 32400$