It's not a counting "method" but a different situation. I'm going to stick with the racing analogy.
You're picking one driver and one car at a time (A1). The other drivers and the other cars are irrelevant. This driver+car does a qualifying lap around the track. You then give them a different car (A2), they lap, then lap again with the third car (A3). Repeat that with the other two drivers (B1-3, C1-3) and you have a total of nine qualifying laps. This way you can see a driver's performance change with respect to which car they were driving (A1-3, B1-3, C1-3), or alternatively how different drivers handle the same car (A-C1, A-C2, A-C3).
grissom is picking all the drivers and all the cars and putting them into a race all at once. Then the assignments change, except rather than have every driver get a different car, what changes is that somebody (perhaps more than one person) gets a different car. Thus the race lineup as a whole changes, even if some or most (but not all) of the drivers stay in the same car they were just driving.
The combinations are
1. A1B2C3
2. A1B3C2
3. A2B1C3
4. A2B3C1
5. A3B1C2
6. A3B2C1
This is more geared towards measuring the performance of a driver with respect to how the other drivers are doing. As is what happens a real race. For example, with races 1 and 2 you can see how driver A (who kept the same car for both) did when B and C changed cars. If you think A's car is the variable, you can compare races 3 and 4 where A used a different car from before (but kept it for both races) and B and C changed. Or races 5 and 6. For driver B you can compare 3+5/1+6/2+4 and for C it's 4+6/2+5/1+3.
tldr: you're counting driver+car combinations, grissom is counting driver+car+driver+car+driver+car combinations.