Permutation & Combination Calculator

In the worlds of probability, statistics, and everyday event planning, we frequently need to figure out how many different ways a set of items can be grouped or arranged. Whether you are an Indian student studying for JEE or CBSE board exams, a sports coach selecting a playing 11 for a cricket match, or a data analyst creating unique passwords, determining the number of possible outcomes is essential.

Our free online Permutation & Combination Calculator instantly computes the number of possible arrangements (nPr) and selections (nCr) from a dataset. You do not need to struggle with complex factorial calculations by hand. The tool automatically processes the exact mathematical formulas to give you clear, instant answers.

Simply tell the calculator your total number of items, how many you want to choose, and whether the order of selection matters.

How to Use the Permutation & Combination Calculator

Using this tool is straightforward. It guides you from raw numbers to a fully evaluated mathematical outcome. Follow these easy steps:

Select Your Calculation Type: At the top left, click the drop-down menu to choose what you need. You can pick:
- Combinations (nCr): Order does NOT matter.
- Permutations (nPr): Order strictly matters.
- Options for calculating "With Replacement" if items can be chosen multiple times.
Enter Total Number of Items (n): This is your complete pool of choices. You can type the number into the input box or use the slider bar. For example, if you have 5 different books, your 'n' is 5.
Enter Number of Items to Choose (r): This is how many items you are selecting from the total pool. If you want to pick 3 books out of the 5, your 'r' is 3.
Review Your Output: Instantly, the large green box on the right will display your final calculated result (e.g., 10 combinations).
Check the Step-by-Step Breakdown: Below the main result, a details table will show you the exact Formula used, the Step-by-Step Calculation, and the Exact Result, making it incredibly helpful for homework and assignments.

Permutations vs. Combinations: What is the difference?

The core difference between the two concepts comes down to one simple question: Does the sequence or order matter?

Permutations: Order does matter. A combination lock is actually a permutation lock! The sequence 1-2-3 will open the lock, but 3-2-1 will not. They are different permutations.
Combinations: Order does not matter. If you are making a fruit salad, adding apples, bananas, and grapes is the exact same salad as adding grapes, bananas, and apples. It is one single combination.

The Mathematical Formulas

In these formulas, n represents the total number of items available, r represents the number of items being chosen, and the exclamation point (!) represents a factorial (multiplying a number by every whole number below it down to 1).

Combinations (nCr)

Used when forming groups, teams, or selecting lottery numbers.

n! / (r! × (n - r)!)

Permutations (nPr)

Used for passwords, seating arrangements, or race finishes.

n! / (n - r)!

What does "With Replacement" mean?

Standard permutations and combinations assume that once you pick an item, you cannot pick it again (like drawing a card from a standard deck). If you replace the item after picking it (like rolling a die multiple times or creating a PIN code where numbers can repeat), you must use the "With Replacement" calculation modes. Permutations with replacement follow the much simpler formula n^r.

Real-Life Worked Examples

Example 1: Forming a Cricket Team (Combination - nCr)

Scenario: You are the coach of an Indian local cricket club. You have a squad of 15 players, but you can only select 11 players for the final starting team. The order in which you pick them does not matter; either they are on the team or they are not.

Total Items (n): 15 (Total players)
Items to Choose (r): 11 (Players needed)
Formula applied: 15! / (11! × (15 - 11)!)
Calculation: 15! / (11! × 4!)

Result: There are 1,365 different combinations of players you can choose to form your final 11.

Example 2: Awarding Medals in a Race (Permutation - nPr)

Scenario: 10 sprinters are competing in a 100m dash at a school sports day. Medals are awarded for Gold (1st place), Silver (2nd place), and Bronze (3rd place). Because the order completely changes who gets which medal, order matters.

Total Items (n): 10 (Total runners)
Items to Choose (r): 3 (Top 3 positions)
Formula applied: 10! / (10 - 3)!
Calculation: 10! / 7!
Step-by-step math: 10 × 9 × 8

Result: There are 720 different ways the top 3 medals can be awarded.

Frequently Asked Questions

What is the main difference between a permutation and a combination?▼

The main difference is whether the order of the items matters. In a permutation, the order matters (e.g., a PIN code where 1234 is different from 4321). In a combination, the order does not matter (e.g., a mixed fruit bowl containing apples and bananas is the same as one containing bananas and apples).

What does the exclamation mark (!) mean in these formulas?▼

The exclamation mark represents a mathematical operation called a 'factorial.' It means you multiply the number by every whole number below it down to 1. For example, 5! (five factorial) is 5 × 4 × 3 × 2 × 1, which equals 120.

Can the number of items chosen (r) be greater than the total items (n)?▼

No, if you are calculating standard combinations or permutations without replacement, 'r' cannot be greater than 'n'. You cannot select 6 items if you only have 5 items available to choose from.

What does 'with replacement' mean in probability?▼

With replacement means that after you select an item, you put it back into the pool of options before making your next selection. This means the same item can be chosen more than once. Setting a 4-digit suitcase lock (where numbers 0-9 can be repeated) is an example of permutations with replacement.

Why is a combination lock actually a permutation lock?▼

It is a funny quirk of the English language! For a 'combination' lock, the sequence of the numbers strictly matters to open it. Since order matters, it is mathematically a permutation, not a combination.

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