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