Combination Calculator: Calculate Permutations and Combinations

Combinations and permutations are fundamental concepts in probability, statistics, and combinatorics. A combination counts selections where order does not matter, while a permutation counts arrangements where order matters. Our Combination Calculator computes both values quickly, along with related metrics like probability and the number of possible subsets.

Understanding Combinations vs Permutations

The key difference is whether order matters. Choosing 3 people from a group of 10 for a committee is a combination — the order of selection does not matter. Choosing 3 people from 10 for president, vice president, and secretary is a permutation — the order matters because each position is different.

Combination formula: nCr = n! / (r! × (n-r)!). Permutation formula: nPr = n! / (n-r)!. For n=10, r=3: 10C3 = 120 combinations, 10P3 = 720 permutations.

Using the Combination Calculator

Enter the total number of items (n) and the number of items to choose (r). Select whether you want combinations (order does not matter) or permutations (order matters). The calculator shows the result, the formula used, and a step-by-step breakdown of the calculation.

You can also choose whether repetition is allowed. With repetition allowed, combination counts increase dramatically because items can be selected multiple times. The calculator handles all four scenarios: combinations with and without repetition, and permutations with and without repetition.

Real-World Applications

Lottery Probabilities

In a typical 6/49 lottery where 6 numbers are drawn from 49: 49C6 = 13,983,816 possible combinations. Your chance of winning the jackpot with one ticket is 1 in 14 million. Understanding combinations helps you make informed decisions about gambling odds.

Card Games

A standard 52-card deck: 52C5 = 2,598,960 possible 5-card poker hands. The probability of being dealt a royal flush is 4 / 2,598,960 = 1 in 649,740. Poker hand probabilities are all derived from combination calculations.

Password Security

If a password requires 8 characters from 26 letters and 10 digits (36 total), the number of possible passwords is 36^8 = 2.8 trillion permutations with repetition. Each additional character multiplies the search space exponentially.

Sports Brackets

A 64-team NCAA tournament bracket has 2^63 possible combinations — more than the number of atoms on Earth. This explains why a perfect bracket has never been verified.

Formulas Reference

• Combinations (no repetition): nCr = n! / (r! (n-r)!)
• Permutations (no repetition): nPr = n! / (n-r)!
• Combinations (with repetition): (n+r-1)C(r) = (n+r-1)! / (r! (n-1)!)
• Permutations (with repetition): n^r
• Factorial: n! = n × (n-1) × (n-2) × ... × 1

Real-World Example

A restaurant offers 8 toppings for pizzas. How many different 3-topping pizzas can you order?

• Combination (order of toppings does not matter): 8C3 = 8! / (3! × 5!) = 56 different pizzas
• What if you can choose the same topping multiple times? (8+3-1)C3 = 10C3 = 120 combinations
• If you are arranging 3 toppings on a pizza in a specific order (first, second, third): 8P3 = 336 permutations

This is why menus often list standard pizza combinations — with 8 toppings, there are 256 possible combinations of any size (including no toppings to all 8 toppings).

Start Calculating

Use our Combination Calculator below to compute permutations and combinations for any problem. Also check our Statistics Calculator and our Percentage Calculator for related probability and statistics tools.