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Show that the number of ways in which the letters of the word "arrange" can be arranged so that the two r's are not consecutive is $900$.

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closed as off-topic by Ian Miller, Namaste, Ken Duna, draks ..., hardmath Mar 14 '17 at 16:47

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  • 3
    $\begingroup$ Possible duplicate of Arrangement of word MISSISSIPPI in which no three S occur together $\endgroup$ – HSN Mar 14 '17 at 9:38
  • $\begingroup$ Hi, welcome to Math.SE. Please indicate what you have tried, your thoughts on the problem and where you got stuck. This will help people better tailor their answer to your background and situation. It will also demonstrate that you are interested in your question and not just looking for someone to do your homework for you - Math.SE is not a homework site. $\endgroup$ – Ian Miller Mar 14 '17 at 9:41
  • $\begingroup$ In the title the word "Arrange" is capitalized, but not in the body of your Question. This might lead to different interpretations of the problem accordingly as the two letters $A,a$ are considered identical or distinguishable. $\endgroup$ – hardmath Mar 14 '17 at 16:46
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  1. Number of arrangements of AANGE is $\frac{5!}{2!}=60$
  2. In every arrangement of AANGE we can select two different positions where to put additional R in ${6 \choose 2}=15$ ways
  3. Number of our arrangemenst of ARRANGE without consecutive Rs is then $60\cdot 15=900$
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Here, Total number of letter(n) = 7 Since letter R is repeated twice and letter A is repeated twice, So the total arrangement of letter "arrange" is(x) =n!/(2!*2!) =7!/(2!*2!) =1260

Now, The arrangement of letter"arrange" when the letter r comes together(m) = 6!/(2!) =360 So, The total arrangement of letter "arrange" when two r's does not come together is = x - m = 1260-360 = 900

Hence The total arrangement is 900 for r not coming together.

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