I am wondering, is it possible to draw ten dots so that every dot has the same distance to every other one?
And how many possibilities are there?
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No: three dots having the same distances to each other form an equilateral triangle, and there is no way to add a fourth to have the same distance to all three of those. It would have to be on all three of the circles in this image:
(So it certainly isn't possible to do 10 dots!)
In general, you need $n-1$ dimensions for $n$ dots with this property: these dots will form the vertices of a regular $k$-simplex.
Instead of ten, the most you can get is three dots all at the same distance from each other.
To see why this is, let's start with two dots $A,B$ a given distance apart (say 1 unit). Where can we place a third dot which is 1 unit away from both of them? This dot must be on both of the radius-1 circles centred at $A$ and $B$, and these circles only intersect in two points:
The two points $C1,C2$ aren't distance 1 apart, so the most we can get is three dots, forming an equilateral triangle in the plane.
A similar argument in 3D space shows that the most you can get is four dots all the same distance apart, forming a regular tetrahedron in 3-space.
As Deusovi correctly says in his answer, the general solution in $n$ dimensions is the vertices of a simplex (the higher-dimensional analogue of a triangle or tetrahedron) - such a shape has $n+1$ vertices, all of which are equidistant from each other, and it's impossible to do better than this. So in a space of dimension 9 or higher, it is possible to find ten points all the same distance apart. But we'd need to ask a string theorist for that ...
In two dimensions, there is
a trivial solution: the dots are identical (i.e. at the same place), the distance is trivially zero.
You never said you want Euclidean solutions :-). If the distance is defined by discrete metric, then the distance between any non-identical dots is trivially 1.
EDIT: answer to how many possibilities there are:
for the trivial solution - one
for the non-Euclidean - infinity. And there are other non-trivial metrics you can use, e.g. Chebyshev distance works as well.