(i) Find how many arrangements can be made with the letters of the word 'MATHEMATICS'.
(ii) In how many of them are the vowels together?

Explanation

To solve the problem step by step, we will break it down into two parts as given in the question.

### Part (i): Find how many arrangements can be made with the letters of the word 'MATHEMATICS'.

1. **Count the total number of letters**:
The word 'MATHEMATICS' has 11 letters.

2. **Identify the repeating letters**:
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