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