Step by step
The question
Question: 10 friends meet.
Each pair shakes hands once.
How many handshakes are there?
1 Order: yes or no
Does order matter?
A shaking hands with B is the same handshake as B with A.
So order does not matter.
That is called a combination.
2 Write the rule
The rule for combinations: n factorial, over r factorial times n minus r, factorial.
n is how many we choose from: 10.
r is how many we pick: 2.
3 Put the numbers in
We put 10 for n and 2 for r.
10 minus 2 is 8.
4 Write out and cancel
We write every factorial out.
8 times 7 times 6 times 5 times 4 times 3 times 2 times 1 is on top and at the bottom.
They cancel.
What is left is 10 × 9, over 2 × 1.
Why divide by 2 factorial?
The top, 10 times 9, counts in order.
One group of 2 can be put in order 2 ways.
So each group is counted 2 times.
Careful!
10 times 9 counts each group 2 times.
We must divide by 2.
5 Work it out
2 times 1 is 2.
10 on top and 2 at the bottom both divide by 2.
What is left is 5 × 9.
5 times 9 is 45.
So there are 45 handshakes.
6 Check
Let's check: 45 groups, each in 2 orders, make 90.
And 10 times 9, in order, is 90.
The same.
So the answer is right!
Let's look back.
First: order: yes or no.
Then: write the rule.
Then: put the numbers in.
Then: write out and cancel.
Then: work it out.
Last: check.