What keywords that I should search for references for this kind of problems - vector comparison? Set partitions?

I have two vectors of 1 by 3 dimension and each cell can only take value 
of either 0 or 1. For example, vector A can be (0,1,1), (1,0,1) or 
(1,1,0) and vector B can be (0,1,1), (1,0,1) or (1,1,0).
My question is that how many ways (permutations?) that vector B is 
greater than vector A?
I need the definition of comparing vectors. In my example above, (1,1,0) 
> (0,1,1) and (1,0,1) > (0,1,1). Therefore, there are two situations that 
vector B > vector A.

Couple you please tell me what keywords (vector comparison, set 
partition, combinatorial, etc) that I should use to search for the 
references of this kind of problem?

Thank you for your help.

John