The number of ways in which the letters of the word ARRANGE be arranged so that
(i) the two R's are never together,
(ii) the two A's are together but not two R's.
(iii) neither two A's nor two R's are together.

Explanation

To solve the problem of arranging the letters of the word "ARRANGE" under the given conditions, we will break it down step by step.

### Step 1: Total Arrangements of the Word "ARRANGE"
The word "ARRANGE" consists of 7 letters where:
- A appears 2 times
- R appears 2 times
- N, G, and E appear 1 time each

...