• # question_answer The number of ways in which the letters of the word ARRANGE can be arranged such that both R do not come together is [MP PET 1993] A) 360 B) 900 C) 1260 D) 1620

The word ARRANGE, has AA, RR, NGE letters, that is two A' s, two R's and N, G, E one each. $\therefore$ The total number of arrangements =$\frac{7\,!}{2\,!\,2\,!\,1\,!\,1\,!\,1\,!}=1260$ But, the number of arrangements in which both RR are together as one unit = $\frac{6\,!}{2\,!\,1\,!\,1\,!\,1\,!\,1\,!}=360$ $\therefore$ The number of arrangements in which both RR do not come together  = 1260 - 360 = 900.