To solve the question step by step, we will break it down into two parts as given.
### Part (i): Find how many arrangements can be made with the letters of the word "MATHEMATIC".
1. **Count the total letters**:
The word "MATHEMATIC" has 11 letters.
2. **Identify the repeating letters**:
In "MATHEMATIC", the letters are:
- M: 2 times
- A: 2 times
- T: 2 times
- H: 1 time
- E: 1 time
- I: 1 time
- C: 1 time
3. **Apply the formula for permutations of multiset**:
The formula for the number of arrangements of letters in a word where some letters are repeated is given by:
\[
\text{Number of arrangements} = \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!}
\]
where \( n \) is the total number of letters, and \( p_1, p_2, \ldots, p_k \) are the frequencies of the repeating letters.
Here, \( n = 11 \) (total letters), and the frequencies are:
- M: 2
- A: 2
- T: 2
Thus, the formula becomes:
\[
\text{Number of arrangements} = \frac{11!}{2! \times 2! \times 2!}
\]
4. **Calculate the factorials**:
- \( 11! = 39916800 \)
- \( 2! = 2 \)
Therefore:
\[
\text{Number of arrangements} = \frac{39916800}{2 \times 2 \times 2} = \frac{39916800}{8} = 4989600
\]
### Part (ii): In how many of them do the vowels occur together?
1. **Identify the vowels in "MATHEMATIC"**:
The vowels are A, A, E, I (4 vowels).
2. **Treat the vowels as a single entity**:
If we treat the group of vowels (AAEI) as a single letter, we can represent the arrangement as:
- (AAEI), M, T, H, M, T, C
This gives us a total of 8 entities to arrange: (AAEI), M, T, H, M, T, C.
3. **Count the repeating letters**:
The letters are:
- M: 2 times
- T: 2 times
- (AAEI): 1 time (considered as one letter)
4. **Apply the permutations formula**:
The number of arrangements of these 8 entities is:
\[
\text{Number of arrangements} = \frac{8!}{2! \times 2!}
\]
5. **Calculate the factorials**:
- \( 8! = 40320 \)
Therefore:
\[
\text{Number of arrangements} = \frac{40320}{2 \times 2} = \frac{40320}{4} = 10080
\]
6. **Arrange the vowels within their group**:
The vowels (AAEI) can be arranged among themselves. The number of arrangements of the vowels is:
\[
\text{Arrangements of vowels} = \frac{4!}{2!} = \frac{24}{2} = 12
\]
7. **Total arrangements with vowels together**:
Finally, multiply the arrangements of the entities by the arrangements of the vowels:
\[
\text{Total arrangements} = 10080 \times 12 = 120960
\]
### Final Answers:
(i) The total arrangements of the letters in "MATHEMATIC" is **4989600**.
(ii) The arrangements where the vowels occur together is **120960**.
