The letters of the word “ARRANGE” are arranged in all possible ways. Let m be the number of arrangements in which the two A's are together and the two R's are not together and n be the number of arrangements in which neither the two A's nor the two R s are together. Then

The number of arrangements in which the two A's are together and the two R's are not together

(AA), N,G,E can be arranged in 4! ways

There are are 5 gaps to arrange 2 R's in ${ }^5{C}_2=10 ways$

No of ways in which As come together but R's do not come together=m=24x10=240

No of ways in which R's come together 

(RR),N,G,E A,Acan be arranged in 6! /2ways 

No of ways in which R's come together =360

No of ways in which A's come together =360

No of ways in which both A's and R's come together in 5!=`120

(RR),N,G,E (A,A)

Total no of ways in which A,A,R,R,N,G,E can be arranged =7!/2!2!=1260

The number of arrangements in which neither the two A's nor the two R s are together. 

n=1260-(360+360-120)=660

$m=240 \mathrm{\&} n=660$