Despite the 2x2 Rubik's Cube being the baby of the family, it's no slouch when it comes to the permutations department.
The corners have eight-factorial positions as well as having three orientations (and three colours). Actually the eighth one doesn't have much choice on where to go if the other seven have taken their positions, so it's really only three to the power of seven possibilities.
Each of the six faces can be rotated through four positions. So the number of possible permutations for the little 2x2 cube is:
Corner Permutations multiplied by the Orientation Permutation all divided by the Rotational Permutations.
(8! x 3^7) / (4 x 6) = 3,674,160
With a travel heavy holiday coming up, I recently challenged myself to be able to solve the 2x2 Rubik's Cube “quickly and efficiently”. The efficiency would come from learning as few algorithms as possible (three or less). As for the quickly part, the 3.6 million possible permutations mean that you could stumble across the solution within a few weeks of constant play purely by manipulating the pieces at random. That being said, the world record is less than half a second. Perhaps something below twenty seconds would be suitable. Ten seconds would be exceptional.
We'll see how things go.