Step by step
The question
Question: A coach picks 4 of 9 players for a team.
How many different teams are there?
1 Order: yes or no
Does order matter?
Players A, B and C are the same team as C, 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: 9.
r is how many we pick: 4.
3 Put the numbers in
We put 9 for n and 4 for r.
9 minus 4 is 5.
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.
What is left is 9 × 8 × 7 × 6, over 4 × 3 × 2 × 1.
Why divide by 4 factorial?
The top, 9 times 8 times 7 times 6, counts in order.
One team of 4 can be put in order 24 ways.
So each team is counted 24 times.
Careful!
9 times 8 times 7 times 6 counts each team 24 times.
We must divide by 24.
5 Work it out
9 times 8 times 7 times 6 is 3024.
4 times 3 times 2 times 1 is 24.
9 times 8 is 72.
72 times 7 is 504.
504 times 6 is 3024.
3024 divided by 24 is 126.
So there are 126 teams.
6 Check
Let's check: 126 groups, each in 24 orders, make 3024.
And 9 times 8 times 7 times 6, in order, is 3024.
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.