In how many distinct arrangements of the letters of 'MATHEMATICS' do the vowels always occur together?

Problem restatement
Arrange the 11 letters of 'MATHEMATICS' so that all vowels remain in one block.

Given data

• Letters: M(2), A(2), T(2), H(1), E(1), I(1), C(1), S(1)
• Vowels: A, A, E, I (4 vowels; A repeats)

Concept/Approach
Treat the 4 vowels as one super-block. Then arrange this block with the 7 consonants around it, accounting for repeated consonants. Multiply by the internal arrangements of the vowels (accounting for repeated A).

Step-by-Step calculation
Consonants: M, M, T, T, H, C, S (7 items with repeats M(2), T(2))Entities to arrange = 7 consonants + 1 vowel-block = 8Ways to arrange these 8 = 8! ÷ (2! 2!) = 40320 ÷ 4 = 10080Vowel-block arrangements (A, A, E, I) = 4! ÷ 2! = 12Total arrangements = 10080 × 12 = 120,960

Verification/Alternative
Check counts of repeats carefully: two M's and two T's in the outer arrangement; two A's inside the block.

Common pitfalls
Ignoring duplicate letters (overcounting), or forgetting to multiply by internal arrangements of the vowel-block.

Final Answer
120,960