# 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

$=1260-360$

$=900$