Step by step
The question
Question: 8 runners run a race.
A bet picks the 1st, 2nd and 3rd places, in order, at random.
What is the probability that the bet wins?
The probability is the number of ways we want, over all the ways.
P stands for probability.
1 Order: yes or no
Does order matter?
Yes.
A first, B second is not B first, A second.
So we count permutations.
2 Count all the ways
8 P 3: 8 choices for the first place, then 7 for the second, then 6.
8 times 7 times 6 is 336 ways.
3 Count the ways we want
Only one order wins: the real 1st, 2nd and 3rd.
So 1 way.
4 Write the fraction
1 way we want, out of 336.
Careful!
Not 8 C 3: that counts groups.
Here the places are different, so order matters.
5 Check
Let's check another way: one place at a time.
The right 1st: 1 out of 8.
Then the right 2nd: 1 out of the 7 left, and 1 out of 6.
Multiplied, that is 1/336.
The same.
So the answer is right!
Let's look back.
First: order: yes or no.
Then: count all the ways.
Then: count the ways we want.
Then: write the fraction.
Last: check.