Different words are being formed by arranging the letters of the word ARRANGE. All the words obtained are written in the form of a dictionary Number of ways (words) in which neither two A's nor two R's comes together

The word ARRANGE has 7 letters: A, A, R, R, N, G, E. To ensure no two A's and no two R's are together, we use the gap method. First, arrange the 3 letters N, G, E in 3! = 6 ways. There are 4 gaps created. We place the 2 A's in 4C2 ways and 2 R's in 2C2 ways. Total = 6 * 6 * 1 * (2!/2!) * (2!/2!) is not correct; the actual calculation involves placing A's and R's into the slots. The correct count is 660.