Multiple combinations from a single set suppose I have a single set of 9 elements {A, B, C, D, E, F, G, H, I} and that I need to create 3 groups of 3 elements each. I know that I can do:
9C3 * 6C3 * 3C3 and that is 1680 different combinations but in those combinations may appear {{A, B, C}, {D, E, F}, {G, H, I}} and {{D, E, F}, {G, H, I}, {A, B, C}}. 
Which is the formula to calculate this? I have the notion that I need to divide by something by don't know by what.

Your reasoning shows that there are $\binom{9}{3}\binom{6}{3}\binom{3}{3}$ ways to divide our people into three teams of three each, one of the teams to wear blue uniforms,one to wear white, and one to wear red. 
Every "nudist camp" division into teams gives rise to $3!$ divisions into uniformed teams. So if $N$ is the number of nudist camp divisions, then: $N=\frac 1{3!}\binom{9}{3}\binom{6}{3}\binom{3}{3}$. 
Remark: Alternately,  line up our people in order of age, or student number, or whatever. The first person can choose the rest of her team in $\binom{8}{2}$ ways. The first unchosen person can then choose the rest of her team in $\binom{5}{2}$ ways. So the number of ways is $\binom{8}{2}\binom{5}{2}$. 
