ARCHITECTURE has 12 letters: A, R, C, H, I, T, E, C, T, U, R, E

Vowels: A, I, E, U, E → 5 vowels (E repeated twice)

Consonants: R, C, H, T, C, T, R → 7 consonants (C×2, T×2, R×2)

Treat all vowels as one unit: we have 8 units (1 vowel-block + 7 consonants).

Arrangements of 8 units with C, T, R each repeated twice: \(\dfrac{8!}{2! \cdot 2! \cdot 2!}\

Arrangements of vowels within the block (5 vowels, E repeated twice): \(\dfrac{5!}{2!}\

Total = \(\dfrac{8!}{2!\cdot2!\cdot2!} \times \dfrac{5!}{2!}\