The number of ways in which the letters of the word ARRANGE can be made such that both R’s do not come together is

Total ways without restricton =_{$\frac{7!}{2!2!}=1260$} Number of ways when R’s come together _{$=A\left[RR\right]ANGE=\frac{6!}{2!}=360$}

_$\therefore$  Required number of ways =1260 -360 =900