Step by step
The question
Question: Maya picks 3 of her 7 books for a trip.
How many ways can she pick?
1 Order: yes or no
Does order matter?
Books A and B make the same pick as books B and A.
So order does not matter.
That is called a combination.
2 Write the rule
The rule for combinations: n factorial, over r factorial times n minus r, factorial.
n is how many we choose from: 7.
r is how many we pick: 3.
3 Put the numbers in
We put 7 for n and 3 for r.
7 minus 3 is 4.
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.
What is left is 7 × 6 × 5, over 3 × 2 × 1.
Why divide by 3 factorial?
The top, 7 times 6 times 5, counts in order.
One group of 3 can be put in order 6 ways.
So each group is counted 6 times.
Careful!
7 times 6 times 5 counts each group 6 times.
We must divide by 6.
5 Work it out
3 times 2 times 1 is 6.
6 on top and 6 at the bottom both divide by 6.
What is left is 7 × 5.
7 times 5 is 35.
So there are 35 ways.
6 Check
Let's check: 35 groups, each in 6 orders, make 210.
And 7 times 6 times 5, in order, 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: work it out.
Last: check.