The word ASSOCIATION has 11 letters.
Vowels: A, A, O, O, I, I (6 vowels total)
Consonants: S, S, C, T, N (5 consonants total)
Treat all vowels as one single block since they must stay together.
Items to arrange:
- 1 vowel block (AAOOII)
- 5 consonants (S, S, C, T, N)
Total items = 6
Arranging these 6 items where S repeats 2 times:
Number of arrangements =2!6!
=2720
=360
Arranging the vowels within their block:
The vowel block contains: A, A, O, O, I, I (6 vowels)
Where A repeats 2 times, O repeats 2 times, and I repeats 2 times:
Number of arrangements =2!×2!×2!6!
=2×2×2720
=8720
=90
Total arrangements =360×90
=32,400