Step by step
The question
Question: work out 9 C 2.
1 Write the rule
The rule for combinations: n factorial, over r factorial times n minus r, factorial.
Here n is 9 and r is 2.
2 Put the numbers in
We put 9 for n and 2 for r.
9 minus 2 is 7.
3 Write out and cancel
We write every factorial out.
7 times 6 times 5 times 4 times 3 times 2 times 1 is on top and at the bottom.
They cancel.
What is left is 9 × 8, over 2 × 1.
Why divide by 2 factorial?
The top, 9 times 8, counts in order.
One group of 2 can be put in order 2 ways.
So each group is counted 2 times.
Careful!
9 times 8 counts each group 2 times.
We must divide by 2.
4 Work it out
2 times 1 is 2.
8 on top and 2 at the bottom both divide by 2.
What is left is 9 × 4.
9 times 4 is 36.
So 9 C 2 is 36.
5 Check
Let's check: 36 groups, each in 2 orders, make 72.
And 9 times 8, in order, is 72.
The same.
So the answer is right!
Let's look back.
First: write the rule.
Then: put the numbers in.
Then: write out and cancel.
Then: work it out.
Last: check.