Question

In how many ways can the letters of the word 'ARRANGE' be arranged so that the two R's are never together?

Answer 1

The word ARRANGE consists of 7 letters including two Rs and two As, which can be arranged in $\frac{7!}{2!2!}$ways.
∴ Total number of words that can be formed using the letters of the word ARRANGE = 1260
Number of words in which the two Rs are always together = Considering both Rs as a single entity
= Arrangements of 6 things of which two are same (two As)
= $\frac{6!}{2!}$
= 360
Number of words in which the two Rs are never together = Total number of words $-$ Number of words in which the two Rs are always together
= 1260$-$360
= 900