The probability of being in the first or second position in a heat with 10 cars can be calculated by finding the total number of ways that the cars can be arranged in the heat and then dividing by the number of ways that result in being in the first or second position.
There are 10 cars in the heat, so there are 10 possible positions for the first car, and once that car has been assigned a position, there are 9 possible positions remaining for the second car, and so on. Therefore, the total number of ways that the cars can be arranged is:
10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
This is equivalent to 10! (read as "10 factorial"), which is the product of all the positive integers from 1 to 10.
To be in the first or second position, there are 2 possible positions out of the 10 total positions in the heat. So, the number of ways to be in the first or second position is:
2 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
which is equivalent to 2 x 9!, since we only need to consider the arrangements of the remaining 9 cars after we have assigned ourselves to one of the first two positions.
Therefore, the probability of being in the first or second position is:
(2 x 9!) / 10!
Simplifying this expression, we get:
(2 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1) / (10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1)
which simplifies further to:
2 / 10
or:
1 / 5
So, the probability of being in the first or second position is 1/5 or 0.2, which means that there is a 20% chance of being in one of those positions in any given heat.