Step by step
The question
Question: work out 10 P 3.
10P3 =
1 Write the rule
The rule for permutations: n factorial, over n minus r, factorial.
nPr = n!(n − r)!
Here n is 10 and r is 3.
n = 10
r = 3
2 Put the numbers in
We put 10 for n and 3 for r.
10P3 = 10!(10 − 3)!
10 minus 3 is 7.
10P3 = 10!7!
Careful!
The bottom is 10 minus 3, factorial.
Not 3 factorial.
Careful: not 10!3! ✗
3 Write out and cancel
We write every factorial out.
10P3 = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 17 × 6 × 5 × 4 × 3 × 2 × 1
7 times 6 times 5 times 4 times 3 times 2 times 1 is on top and at the bottom.
They cancel.
4 Multiply
What is left is 10 × 9 × 8.
10P3 = 10 × 9 × 8
10 times 9 is 90.
90 times 8 is 720.
10 × 9 = 90
90 × 8 = 720
So 10 P 3 is 720.
10P3 = 720
5 Check
10 times 9 times 8 is 720.
The same.
So the answer is right!
10 × 9 × 8 = 720 ✓
Let's look back.
First: write the rule.
Then: put the numbers in.
Then: write out and cancel.
Then: multiply.
Last: check.
Answer
10P3 = 720