Step by step
The question
Question: work out 8 C 3.
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 3.
2 Put the numbers in
We put 8 for n and 3 for r.
8 minus 3 is 5.
3 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 8 × 7 × 6, over 3 × 2 × 1.
Why divide by 3 factorial?
The top, 8 times 7 times 6, counts in order.
One group of 3 can be put in order 6 ways.
So each group is counted 6 times.
Careful!
8 times 7 times 6 counts each group 6 times.
We must divide by 6.
4 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 8 × 7.
8 times 7 is 56.
So 8 C 3 is 56.
5 Check
Let's check: 56 groups, each in 6 orders, make 336.
And 8 times 7 times 6, in order, is 336.
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.