Find the number of different 8-letter arrangements that can be made from the letters of the word DAUGHTER so that(i) all vowels occur together    (ii) all vowels do not occur together.

Explanation

To solve the problem of finding the number of different 8-letter arrangements that can be made from the letters of the word "DAUGHTER" with the conditions that (i) all vowels occur together and (ii) all vowels do not occur together, we can follow these steps:

### Step 1: Identify the vowels and consonants
In the word "DAUGHTER", the vowels are A, U, and E. The consonants are D, G, H, T, and R.

- **Vowels**: A, U, E (3 vowels)
- **Consonants**: D, G, H, T, R (5 consonants)

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