Step by step
The question
Question: A club has 9 members.
It picks a president, a vice president and a treasurer.
How many ways can it do this?
1 Order: yes or no
Does order matter?
A as president and B as vice president is not the same as the other way round.
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: 9.
r is how many we put in order: 3.
3 Put the numbers in
We put 9 for n and 3 for r.
9 minus 3 is 6.
Careful!
The bottom is 9 minus 3, factorial.
Not 3 factorial.
4 Write out and cancel
We write every factorial out.
6 times 5 times 4 times 3 times 2 times 1 is on top and at the bottom.
They cancel.
5 Multiply
What is left is 9 × 8 × 7.
9 times 8 is 72.
72 times 7 is 504.
So there are 504 ways.
6 Check
9 times 8 times 7 is 504.
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.