Question

In how many ways can the letters of the word MATHEMATICS be arranged so that all the vowels always come together?

• A. 11
• B. 4989600
• C. 20660
• D. 120960
• E. 120880

Hint:

When certain letters must be together, treat them as a single unit. Then arrange the units and the internal arrangement separately.

Answer 1

The Correct Option is D

Step 1: Word MATHEMATICS has letters: M(2), A(2), T(2), H(1), E(1), I(1), C(1), S(1). Total 11 letters.
Step 2: Vowels: A, A, E, I (4 vowels). Treat them as a single unit. So we have: [AAEI] + M, M, T, T, H, C, S = 1 + 7 = 8 units.
Step 3: Arrangements of 8 units with repetitions: M(2), T(2) → $\frac{8!}{2!2!} = \frac{40320}{4} = 10080$.
Step 4: Arrangements within the vowel unit: 4 letters with A repeated twice → $\frac{4!}{2!} = \frac{24}{2} = 12$.
Step 5: Total arrangements = $10080 \times 12 = 120960$.
Step 6: Final Answer: 120960.