Step by step
The question
Question: work out 7 C 2.
1 Write the rule
The rule for combinations: n factorial, over r factorial times n minus r, factorial.
Here n is 7 and r is 2.
2 Put the numbers in
We put 7 for n and 2 for r.
7 minus 2 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 7 × 6, over 2 × 1.
Why divide by 2 factorial?
The top, 7 times 6, counts in order.
One group of 2 can be put in order 2 ways.
So each group is counted 2 times.
Careful!
7 times 6 counts each group 2 times.
We must divide by 2.
4 Work it out
2 times 1 is 2.
6 on top and 2 at the bottom both divide by 2.
What is left is 7 × 3.
7 times 3 is 21.
So 7 C 2 is 21.
5 Check
Let's check: 21 groups, each in 2 orders, make 42.
And 7 times 6, in order, is 42.
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.