Question: How many different possible structures (isomers) are there for the complex,
$\ce{[Mn(OH2)3(NH3)3]^{3+}}$?

I really don't know how to begin with this. I know the answer is 2, but i don't have the faintest idea how this is the case. Since $\ce{Mn}$ has a charge of $3+$, I know that its valence electron configuration is $3d^4$. I also know that $\ce{NH3}$ and $\ce{H2O}$ are monodentate, which means they contribute 1 lone pair.

Answer: 
Metal complexes with three each of two types of ligands $(\ce{MX3Y3})$ have two isomers: [*mer*- and *fac*-isomers](http://en.wikipedia.org/wiki/Fac%E2%80%93mer_isomerism#Cis.E2.80.93trans_isomerism_and_facial.E2.80.93meridional_isomerism). For example, for $\ce{Co(NH3)3Cl3}$ there are two isomers:

> 
> *Fac* for facial, in which all three ligands of each type are clustered on a single face of the octahedron and the three other three ligands are on the opposite face.
> 
> 
> 

![enter image description here](https://i.stack.imgur.com/eHTK9.png)

> 
> *Mer* for meridional, in which the three ligands of each type are arranged meridianally in a row around the metal and the other three ligands are arranged likewise on the opposite side in a perpendicular meridian.
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> 
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![enter image description here](https://i.stack.imgur.com/0aLUS.png)
The compound you talked about is of the type $MA\_3B\_3$. I will illustrate a similar example here.
There shall be 2 geometrical isomers, '***FACIAL***' and '***MERIDONAL***'

![enter image description here](https://i.stack.imgur.com/OfenJ.jpg)

> 
> To identify which is which, remember that in *MERIDONAL* maximum bond
>  angles between the groups is **180** whereas in facial it is lesser
>  than that. Also **none** of them is optically active as both have a
>  Plane of symmetry (*merdional* has *diagonal POS whereas facial has
>  horizontal*).
> 
> 
> 

Therefore the compound has total **2 STRUCTURAL ISOMERS** (*2 GEOMETRICAL + 0 OPTICAL*).