Step by step
The question
Question: 6 runners are in a race.
How many ways can gold and silver 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: 6.
r is how many we put in order: 2.
3 Put the numbers in
We put 6 for n and 2 for r.
6 minus 2 is 4.
Careful!
The bottom is 6 minus 2, factorial.
Not 2 factorial.
4 Write out and cancel
We write every factorial out.
4 times 3 times 2 times 1 is on top and at the bottom.
They cancel.
5 Multiply
What is left is 6 × 5.
6 times 5 is 30.
So there are 30 ways.
6 Check
6 times 5 is 30.
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.