Key takeaways
- nPr = n! ÷ (n − r)! counts ordered arrangements of r items from n.
- Shortcut: multiply r numbers counting down from n, for example 8P3 = 8 × 7 × 6.
- Use nPr when order matters and nCr when it does not.
- nPr = nCr × r!, and nPn = n!.
In this guide
What is a permutation?
Eight runners are in a race. How many ways can the gold, silver and bronze medals be given out? Winning gold is different from winning silver, so the order matters. Each different ordered result is a permutation.
The notation nPr means the number of permutations of r items taken from n.
Why do permutations matter?
Whenever positions, ranks or sequences are involved, you need permutations. They tell you how many possible results there are, which is the first step in working out a probability or checking how secure a code is.
Choosing a president, secretary and treasurer from 10 people gives 720 possibilities. Choosing a committee of 3 from the same 10 gives only 120. The difference is whether the roles matter.
The nPr formula
| Symbol | What it means | Example |
|---|---|---|
| n | Total items available | 8 runners |
| r | How many you arrange (0 to n) | 3 medals |
| n! | Factorial of n | 8! = 40,320 |
| (n − r)! | Cancels the items not used | 5! = 120 |
| nPr | Number of ordered arrangements | 336 |
An easier way to see it: nPr is the product of r numbers counting down from n. There are 8 choices for gold, then 7 left for silver, then 6 left for bronze: 8 × 7 × 6 = 336.
How to calculate nPr step by step
- 1Write down n and r. Check that r is between 0 and n.
- 2Start at n and count down r numbers. For 5P2, that is 5 and 4.
- 3Multiply them.
5 × 4 = 20, so 5P2 = 20. - 4Check with the full formula if you like.
5! ÷ 3! = 120 ÷ 6 = 20.
Now enter your own n and r in the calculator below and check your answer.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Letting r be bigger than n. You cannot arrange 6 items out of 4.
- Allowing repeats. nPr assumes each item is used at most once. A 4-digit PIN where digits can repeat has
10 × 10 × 10 × 10 = 10,000options, not 10P4 = 5,040. - Counting down too far. 8P3 has exactly 3 factors: 8, 7 and 6. Stop there.
Tips and tricks
- nPn = n!. Arranging all n items is simply n factorial.
- nP1 = n. Picking one item for one spot can be done n ways.
- Quick size check. nPr is always at least as big as nCr, because each group can be ordered in r! ways.
Where you'll use it in real life
- Sport: podium results and league rankings.
- Clubs and classes: choosing a president, secretary and treasurer.
- Seating plans: the ways to seat 4 guests in 4 chairs is 4P4 = 24.
- Probability: counting outcomes for the probability calculator.
Quick summary
Permutations count ordered arrangements: nPr = n! ÷ (n − r)!, or simply multiply r numbers counting down from n. If order does not matter, switch to nCr. Use the calculator above to find any nPr with every step shown.
Worked example
Eight runners enter a school race. Medals go to first, second and third place. How many different ways can the three medals be awarded?
- 01nPr = n! / (n - r)!
- 028P3 = 8! / 5!
- 038P3 = 8 × 7 × 6
- 048P3 = 336
These numbers are pre-filled in the calculator above.
Frequently asked questions
What is the nPr formula?
nPr = n! ÷ (n − r)!. It is the same as multiplying r numbers counting down from n, for example 8P3 = 8 × 7 × 6.
What is the difference between permutation and combination?
In a permutation the order matters, so ABC and CBA are different. In a combination the order does not matter, so they are the same group. nPr = nCr × r!.
What is 10P3?
10P3 = 10 × 9 × 8 = 720.
What is nP0 and nPn?
nP0 = 1, because there is one way to arrange nothing. nPn = n!, because you are arranging all n items.
