Step by step
The question
Question: 8 runners are in a race.
How many ways can gold, silver and bronze be given?
1 Order: yes or no
Does order matter?
Gold for runner A and silver for runner B is not the same as gold for B and silver for A.
So order matters.
That is called a permutation.
2 Write the rule
The rule for permutations: n factorial, over n minus r, factorial.
n is how many we choose from: 8.
r is how many we put in order: 3.
3 Put the numbers in
We put 8 for n and 3 for r.
8 minus 3 is 5.
Careful!
The bottom is 8 minus 3, factorial.
Not 3 factorial.
4 Write out and cancel
We write every factorial out.
5 times 4 times 3 times 2 times 1 is on top and at the bottom.
They cancel.
5 Multiply
What is left is 8 × 7 × 6.
8 times 7 is 56.
56 times 6 is 336.
So there are 336 ways.
6 Check
8 times 7 times 6 is 336.
The same.
So the answer is right!
Let's look back.
First: order: yes or no.
Then: write the rule.
Then: put the numbers in.
Then: write out and cancel.
Then: multiply.
Last: check.