Step by step
The question
Question: How many ways can 3 of 7 books be put in a row on a shelf?
1 Order: yes or no
Does order matter?
Book A then book B is a different row from book B then book 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: 7.
r is how many we put in order: 3.
3 Put the numbers in
We put 7 for n and 3 for r.
7 minus 3 is 4.
Careful!
The bottom is 7 minus 3, factorial.
Not 3 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 7 × 6 × 5.
7 times 6 is 42.
42 times 5 is 210.
So there are 210 ways.
6 Check
7 times 6 times 5 is 210.
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.