Step by step
The question
Question: work out 8 C 2.
1 Write the rule
The rule for combinations: n factorial, over r factorial times n minus r, factorial.
Here n is 8 and r is 2.
2 Put the numbers in
We put 8 for n and 2 for r.
8 minus 2 is 6.
3 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.
What is left is 8 × 7, over 2 × 1.
Why divide by 2 factorial?
The top, 8 times 7, counts in order.
One group of 2 can be put in order 2 ways.
So each group is counted 2 times.
Careful!
8 times 7 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 4 × 7.
4 times 7 is 28.
So 8 C 2 is 28.
5 Check
Let's check: 28 groups, each in 2 orders, make 56.
And 8 times 7, in order, is 56.
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.