please help with quick counting problem

 

   Find the number of ways of arranging the letters of the word 'ARRANGE' so that the two 'R's are not next to each other.     

 

   The total number of ways they can be arranged is ₇P₇ which is 5040. 

   But we want to eliminate the instances wherer the R's are together.  

   The two R's could be in six positions, namely   R R _ _ _ _ _     

                                                                             _ R R _ _ _ _     

                                                                             _ _ R R _ _ _     

                                                                             _ _ _ R R _ _     

                                                                             _ _ _ _ R R _     

                                                                             _ _ _ _ _ R R     

 

   So, let's subtract those six and obtain a total of (5040 – 6) = 5034     

   _.