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

The letters of word ARRANGE can be rewritten as
A R N G E
A R
So we have 2 A's and 2 R's , and total 7 letters.
(i) Total number of words is `(7!)/(2!2!)=1260`.
The number of words in which 2 R's are together [consider (R R) as one unit] is `6!//2!`. e.g.,
The number of words in which 2 R's are together [consider (R R) as one unit] is `6!//2!`. e.g.,
(R R),A,A,N,G,E
Note that permutations of R R give nothing extra. Therefore, the number of words in which the two R's are never together is
`(7!)/(2!2!)-(6!)/(2!)=900`
(ii) The number of words in which both A's are together is `6!//2!=360`, e.g.,
(A A),R,R,N,G,E
The number of words in which both A's and both R's are together is 5!=120, e.g.,
(A A), (R R), N,G,E
Therefore, the number of words in which both A's are together but the two R's are not together is 360-120=240.
(iii) There are in all 900 words in each of which the two R's are never together. Consider any such word. Either the two A's are together or the two A's not together. But the number of all such arrangements in which the two A's are together is 240. Hence, the number of all such arrangements in which the two A's not together is 900-240=660.