Permutation and Combination Calculator: nPr and nCr Explained

How many ways can you arrange five runners on a podium? How many different three-person committees can you form from a group of ten? These are counting problems, and they are solved with permutations and combinations. A permutation and combination calculator gives you nPr and nCr instantly, but understanding the difference between them is the key to using them correctly.

Permutations vs Combinations: The Key Difference

The single most important question is: does order matter?

• Permutations count arrangements where order matters. First, second, and third place on a podium are different outcomes.
• Combinations count selections where order does not matter. A committee of Alice, Bob, and Carol is the same committee no matter what order you name them in.

Get this distinction right and the rest is just plugging numbers into a formula.

The Permutation Formula (nPr)

The number of ways to arrange r items chosen from n distinct items is:

nPr = n! ÷ (n − r)!

A Worked Example

In how many ways can gold, silver, and bronze medals be awarded among 8 athletes?

• nPr = 8P3 = 8! ÷ (8 − 3)! = 8! ÷ 5! = 8 × 7 × 6 = 336.

Order matters here — who gets gold versus bronze is a different outcome — so we use a permutation.

The Combination Formula (nCr)

The number of ways to choose r items from n distinct items when order does not matter is:

nCr = n! ÷ (r! × (n − r)!)

A Worked Example

How many 3-person committees can be formed from 8 people?

• nCr = 8C3 = 8! ÷ (3! × 5!) = 336 ÷ 6 = 56.

Notice this is exactly the permutation answer (336) divided by 3! (the number of ways to order each group of 3). That relationship — nCr = nPr ÷ r! — is worth remembering.

How to Use the Calculator

1. Enter n, the total number of items.
2. Enter r, the number you are choosing or arranging (with 0 ≤ r ≤ n).
3. The calculator displays both nPr and nCr, using exact arithmetic so large values stay precise.

Where Permutations and Combinations Are Used

• Lotteries and games: the odds of winning depend on combinations.
• Passwords and PINs: the number of possible codes is a permutation problem.
• Committees and teams: selecting groups uses combinations.
• Probability: many probability calculations start by counting favourable and total outcomes.

Common Mistakes to Avoid

• Confusing the two. Always ask whether order matters. If it does, use nPr; if not, use nCr.
• Letting r exceed n. You cannot choose more items than you have, so r must be no greater than n.
• Forgetting 0! = 1. This is needed when r equals n.
• Double-counting. Using a permutation when order does not matter overcounts every group by r!.

A Quick Reference

• nP0 = 1 and nC0 = 1 (one way to choose nothing).
• nPn = n! (arrange everything) and nCn = 1 (one way to take everything).
• nCr = nC(n − r) — choosing r to keep is the same as choosing n − r to leave out.

Conclusion

Permutations and combinations are the foundation of combinatorics and probability. Once you can tell whether order matters, the nPr and nCr formulas do the rest. A permutation and combination calculator handles the factorial arithmetic for you and returns both answers at once.

Try our free Permutation and Combination Calculator for instant results.