1. **Problem Statement:** We want to find the probability that the vowels in the word \textbf{MATHEMATICS} appear in alphabetical order (A, A, E, I) when the letters are arranged randomly. 2. **Identify the letters:** The word \textbf{MATHEMATICS} has 11 letters in total. - Vowels: A, A, E, I (4 vowels) - Consonants: M, T, H, M, T, C, S (7 consonants) 3. **Total number of arrangements:** Since there are repeated letters (2 M's and 2 T's, and 2 A's), the total number of distinct arrangements is given by the multinomial formula: $$\text{Total arrangements} = \frac{11!}{2! \times 2! \times 2!}$$ Here, the denominators correspond to the repetitions of A, M, and T. 4. **Condition for vowels in alphabetical order:** The vowels must appear in the order A, A, E, I. This means among the 4 vowel positions, the vowels are fixed in that order. 5. **Number of favorable arrangements:** - First, choose the positions of the 4 vowels out of 11 letters: $$\binom{11}{4}$$ ways. - Since the vowels must be in alphabetical order, there is only 1 way to arrange them in those chosen positions. - The consonants (7 letters with 2 M's and 2 T's) can be arranged in $$\frac{7!}{2! \times 2!}$$ ways. 6. **Calculate the number of favorable arrangements:** $$\text{Favorable} = \binom{11}{4} \times 1 \times \frac{7!}{2! \times 2!}$$ 7. **Calculate the probability:** $$\text{Probability} = \frac{\text{Favorable}}{\text{Total arrangements}} = \frac{\binom{11}{4} \times \frac{7!}{2! \times 2!}}{\frac{11!}{2! \times 2! \times 2!}}$$ 8. **Simplify the expression:** Recall that $$\binom{11}{4} = \frac{11!}{4! \times 7!}$$ Substitute: $$\text{Probability} = \frac{\frac{11!}{4! \times 7!} \times \frac{7!}{2! \times 2!}}{\frac{11!}{2! \times 2! \times 2!}} = \frac{11!}{4! \times 7!} \times \frac{7!}{2! \times 2!} \times \frac{2! \times 2! \times 2!}{11!}$$ Canceling $$11!$$ and $$7!$$: $$= \frac{1}{4!} \times \frac{1}{2! \times 2!} \times 2! \times 2! \times 2! = \frac{2!}{4!} = \frac{2}{24} = \frac{1}{12}$$ 9. **Final answer:** The probability that the vowels appear in alphabetical order is $$\boxed{\frac{1}{12}}$$. 10. **Check options:** None of the options A, B, C, D match \(\frac{1}{12}\). However, the problem likely expects the answer in terms of factorial denominators. Since \(12 = 2 \times 6\), and none of the options match, the closest is none. The correct probability is \(\frac{1}{12}\).