Step by step
The question
Question: work out 9 P 4.
9P4 =
1 Write the rule
The rule for permutations: n factorial, over n minus r, factorial.
nPr = n!(n − r)!
Here n is 9 and r is 4.
n = 9
r = 4
2 Put the numbers in
We put 9 for n and 4 for r.
9P4 = 9!(9 − 4)!
9 minus 4 is 5.
9P4 = 9!5!
Careful!
The bottom is 9 minus 4, factorial.
Not 4 factorial.
Careful: not 9!4! ✗
3 Write out and cancel
We write every factorial out.
9P4 = 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 15 × 4 × 3 × 2 × 1
5 times 4 times 3 times 2 times 1 is on top and at the bottom.
They cancel.
4 Multiply
What is left is 9 × 8 × 7 × 6.
9P4 = 9 × 8 × 7 × 6
9 times 8 is 72.
72 times 7 is 504.
504 times 6 is 3024.
9 × 8 = 72
72 × 7 = 504
504 × 6 = 3024
So 9 P 4 is 3024.
9P4 = 3024
5 Check
9 times 8 times 7 times 6 is 3024.
The same.
So the answer is right!
9 × 8 × 7 × 6 = 3024 ✓
Let's look back.
First: write the rule.
Then: put the numbers in.
Then: write out and cancel.
Then: multiply.
Last: check.
Answer
9P4 = 3024