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Permutations and Combinations, in Plain Words
Suppose you have ten runners and you want to award gold, silver and bronze medals. The order matters: giving gold to Ana and silver to Ben is a different result from giving gold to Ben and silver to Ana. That is a permutation. Now suppose you want to pick any three of the ten runners for a team, and it does not matter who is listed first. That is a combination. The same numbers, n = 10 and r = 3, give different answers: 720 permutations but only 120 combinations.
The difference is the whole subject. Permutations count ordered selections. Combinations count unordered selections. Every combination of three runners can be ordered in 3! = 6 ways, which is exactly why 720 divided by 6 gives 120. Once you see that link, the two formulas stop being two separate facts to memorise and become one idea seen from two sides.
How to Use This Calculator
Enter n, the total number of items, and r, the number you choose. Press Calculate nPr and nCr. The tool shows both answers, plus the factorial steps behind each one. For n = 10 and r = 3, you will see nPr = 10! / 7! = 720 and nCr = 10! / (3! × 7!) = 120. Seeing the steps matters, because the steps are where most errors happen: people divide by the wrong factorial, or forget one entirely.
Both n and r must be whole numbers with 0 ≤ r ≤ n ≤ 100. Choosing more items than exist makes no sense, and the tool will tell you so rather than invent an answer.
A Worked Example: Passwords and Teams
Take a four-digit PIN made from the digits 0 to 9, with no repeated digit. Order matters, because 1234 is not the same PIN as 4321. That is a permutation: 10P4 = 10! / 6! = 5,040 possible PINs. Now take a lottery where you pick six balls from 49 and the order of the balls does not matter. That is a combination: 49C6 = 13,983,816. The lottery number is huge, which is why winning is so unlikely, even though both problems start from the same kind of counting.
Try both in the tool. First n = 10, r = 4, and note the permutation answer. Then n = 49, r = 6, and note the combination answer. The gap between the two answers for the same style of problem is the price of caring about order.
The Formulas, Explained
The permutation formula is nPr = n! / (n − r)!. The reasoning is simple: you have n choices for the first pick, n − 1 for the second, and so on down to n − r + 1 for the last pick. Multiplying those choices gives a chain that is exactly the top of n! with the bottom cut off, which is what dividing by (n − r)! does.
The combination formula adds one more division: nCr = n! / (r! × (n − r)!). That extra r! removes the orderings. Each unordered group of r items can be arranged in r! ways, so dividing by r! collapses all those arrangements into one group. That is the only difference between the two formulas, and it is the difference that matters.
Common Mistakes
The classic mistake is using a permutation when the problem asks for a combination, or the reverse. Ask one question before you calculate: would swapping two chosen items create a different outcome? If yes, use nPr. If no, use nCr. A committee is a combination; a president, vice-president and secretary drawn from the same group is a permutation, because the roles differ.
Another mistake is forgetting that repetition changes everything. These formulas assume each item is used at most once. If digits can repeat in a PIN, the count is 10^4 = 10,000, not 10P4. Always check whether repeats are allowed before you reach for nPr or nCr.
When Each One Fits
Use permutations for passwords, race results, seating orders with named seats, and any ranking. Use combinations for teams, committees, lottery draws, card hands, and any group where only membership matters. If you are unsure, work a tiny case by hand: three items, choose two, and list the results. If your list has 6 entries, order matters. If it has 3, it does not. Then scale up with the tool.
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