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" such that (i) all vowels occur together and (ii) all vowels do not occur together, we can follow these steps:

### Step 1: Identify the Letters and Vowels
The word "DAUGHTER" consists of 8 letters: D, A, U, G, H, T, E, R.
The vowels in this word are A, U, and E.

### Step 2: Calculate Arrangements with All Vowels Together
To find the arrangements where all vowels occur together, we can treat the vowels (A, U, E) as a single entity or "block".
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