In how many ways can the letters of the word ARRANGE be arranged so that the two R's are not together?

ARRANGE has 7 letters with A repeated twice and R repeated twice: total arrangements = 7!/(2! 2!) = 1260.

Arrangements with the two R's together: treat RR as one block → 6 units (block, A, A, N, G, E) → 6!/2! = 360.

Required = 1260 − 360 = 900.

Alternative (gap method): arrange A, A, N, G, E in 5!/2! = 60 ways; this creates 6 gaps, choose 2 for the R's: C(6,2) = 15; total 60 × 15 = 900 ✓.