1. ## permutation question

Using the letters GOOGOOGAGAGA determine the number of distinguishable arrangements with the first O after the first G but not necessarily right after?

G=5
0=4
A=3

I know the total possibilities are 12!/(5!4!3!) but that would include an O coming before a G.

2. Hello, jloco1991!

I think I have approach to this problem.
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 are four cases to consider.

(1) The first $\displaystyle \,G$ is the first letter: .$\displaystyle G\,\+\,\_\,\_\,\_\,\hdots$

. . .The other 11 letters $\displaystyle \{G,G,G,G,O,O,O,O,A,A,A\}$

. . . . .can be arranged in: .$\displaystyle \displaystyle{11\choose4,4,3} \:=\:11,\!550\text{ ways.}$

(2) The first $\displaystyle \,G$ is the second letter: .$\displaystyle A\,G\,\_\,\_\,\_\,\hdots$

. . .The other 10 letters $\displaystyle \{G,G,G,G,O,O,O,O,A,A\}$

. . . . .can be arranged in $\displaystyle \displaystyle{10\choose4,4,2} \:=\: 3,\!150\text{ ways.}$

(3) The first $\displaystyle \,G$ is the third letter: .$\displaystyle A\,A\,G\,\_\,\_\,\_\,\hdots$

. . .The other 9 letters $\displaystyle \{G,G,G,G,O,O,O,O,A\}$

. . . . . can be arranged in $\displaystyle \displaystyle{9\choose4.4.1} \:=\:630\text{ ways.}$

(4) The first $\displaystyle \,G$ is the fourth letter: .$\displaystyle A\,A\,A\,G\,\_\,\_\,\_\,\hdots$

. . .The other 8 letters $\displaystyle \{G,G,G,G,O,O,O,O\}$

. . . . . can be arranged in $\displaystyle \displaystyle{8\choose4,4} \:=\:70\text{ ways.}$

Therefore, there are:

. . $\displaystyle 11,\!550 + 3,\!150 + 630 + 70 \:=\:15,\!400\text{ arrangements.}$