vectors in 3D space and Right-Hand Rule

Question

Suppose we have three vectors in 3D space. My questions are:

1. How we check if these vectors are satisfy the right-hand rule or not. I know that it's possible to make the three vectors satisfy the right-hand rule by either change the signs of one vector, or of all three, or swap two vectors. But
2. How many possible combinations of the three vectors (swapping or change the sign) that makes them satisfy the right-hand rule. and depends on what the vectors are swapped or the signs changes.

Answer 1

I understand your question as follows: How many triples from the set $\{+e_1,-e_1,+e_2,-e_2,+e_3,-e_3\}$ form a basis which is positively oriented with respect to the standard basis $(e_1,e_2,e_3)$ of ${\mathbb R}^3$?

Any such triple has to contain one vector from each pair $\{+e_i,-e_i\}$. As the order of the numbers $i$ is arbitrary, as is the sign chosen in each pair, there are $6\cdot 8=48$ triples forming a basis, and (by symmetry) $24$ of them are positively oriented.