Step by step
The question
Question: work out 9 C 3.
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 3.
2 Put the numbers in
We put 9 for n and 3 for r.
9 minus 3 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 9 × 8 × 7, over 3 × 2 × 1.
Why divide by 3 factorial?
The top, 9 times 8 times 7, counts in order.
One group of 3 can be put in order 6 ways.
So each group is counted 6 times.
Careful!
9 times 8 times 7 counts each group 6 times.
We must divide by 6.
4 Work it out
9 times 8 times 7 is 504.
3 times 2 times 1 is 6.
9 times 8 is 72.
72 times 7 is 504.
504 divided by 6 is 84.
So 9 C 3 is 84.
5 Check
Let's check: 84 groups, each in 6 orders, make 504.
And 9 times 8 times 7, in order, is 504.
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.