Answer:
For each runner we have two selections:
the corrals where they start, we have 10 of them.
The color of the shirt, we have 3 colours.
Then the possible outcomes for each runner is the product between those two number of options:
Outcomes for each runner = 3*10 = 30
Now, if we have 10 different runners. such that we assign one of them to each one of the corrals, we have that in the first corral we can put 1 of the 10 runners.
In the second corral we can put 1 of the 9 runners left (9 because we already take one)
For the third one we need to select between 8 runners, etc
And so on.
Then the number of possible outcomes is:
C = 10*9*8*7*6*5*4*3*2*1
And remember that for each one of these 10 runners, we have 3 possible colours of shirts, so we need to write this as:
C = (10*3)*(9*3)*(8*3)(7*3)*(6*3)*(5*3)*(4*3)*(3*3)*(2*3)*(1*3)
C = 10!*(3^10) = 214,277,011,200
So in this case we have a lot of possible combinations.