Step by step
The question
Question: work out 8 C 4.
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 4.
2 Put the numbers in
We put 8 for n and 4 for r.
8 minus 4 is 4.
3 Write out and cancel
We write every factorial out.
4 times 3 times 2 times 1 is on top and at the bottom.
They cancel.
What is left is 8 × 7 × 6 × 5, over 4 × 3 × 2 × 1.
Why divide by 4 factorial?
The top, 8 times 7 times 6 times 5, counts in order.
One group of 4 can be put in order 24 ways.
So each group is counted 24 times.
Careful!
8 times 7 times 6 times 5 counts each group 24 times.
We must divide by 24.
4 Work it out
8 times 7 times 6 times 5 is 1680.
4 times 3 times 2 times 1 is 24.
8 times 7 is 56.
56 times 6 is 336.
336 times 5 is 1680.
1680 divided by 24 is 70.
So 8 C 4 is 70.
5 Check
Let's check: 70 groups, each in 24 orders, make 1680.
And 8 times 7 times 6 times 5, in order, is 1680.
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.