In how many ways can the letters of the word 'ARRANGE' be arranged so that the two R's are never together?
Question:
In how many ways can the letters of the word 'ARRANGE' be arranged so that the two R's are never together?
Solution:
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
$=1260-360$
$=900$