Hello, jloco1991!

II have approach to this problem.think

If I'm wrong, someone will certainly say so.

Using the letters GOOGOOGAGAGA determine the number of distinguishable

arrangements with the first O after the first G, but not necessarily right after?

. . (Five G's, four O's, three A's)

I see that there arecases to consider.four

(1) The first is theletter: .first

. . .The other 11 letters

. . . . .can be arranged in: .

(2) The first is theletter: .second

. . .The other 10 letters

. . . . .can be arranged in

(3) The first is theletter: .third

. . .The other 9 letters

. . . . . can be arranged in

(4) The first is theletter: .fourth

. . .The other 8 letters

. . . . . can be arranged in

Therefore, there are:

. .