Step by step
The question
Question: work out 10 C 3.
10 C 3 counts the ways to pick 3 of 10 things, when order does not matter.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Write the rule
The rule for combinations: n factorial, over r factorial times n minus r, factorial.
It is the rule for order, with one more bottom, r factorial.
We will see why.
Here n is 10 and r is 3.
2 Put the numbers in
We put 10 for n and 3 for r.
10 minus 3 is 7.
3 Write out and cancel
We write every factorial out.
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 × 8, over 3 × 2 × 1.
Why divide by 3 factorial?
The top, 10 times 9 times 8, counts in order.
One group of 3 can be put in order 6 ways.
So each group is counted 6 times.
Careful!
10 times 9 times 8 counts each group 6 times.
We must divide by 6.
4 Work it out
10 times 9 times 8 is 720.
3 times 2 times 1 is 6.
10 times 9 is 90.
90 times 8 is 720.
720 divided by 6 is 120.
So 10 C 3 is 120.
5 Check
Let's check: 120 groups, each in 6 orders, make 720.
And 10 times 9 times 8, in order, is 720.
The same.
So the answer is right!
6 Pick the choice
Our answer is ₁₀C₃ = 120.
That is choice B.
7 Why not the others
A is 30.
Wrong: 10 × 3: multiplied the numbers.
C is 720.
Wrong: that is 10P3: in a group the order does not matter.
That is the Careful mistake from before.
D is 1000.
Wrong: 10³: the same student cannot be picked twice.
Let's look back.
First: write the rule.
Then: put the numbers in.
Then: write out and cancel.
Then: work it out.
Then: check.
Then: pick the choice.
Last: why not the others.