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. Use our Combination & Permutation Calculator to estimate your costs.
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.
Working Through the Factorials by Hand
To really understand where the numbers come from, expand the factorial terms. Take n = 10, r = 3. The full factorial 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800, but the formula collapses because most of the terms cancel. For combinations, nCr = n! / (r! (n-r)!) = 10! / (3! × 7!) = (10 × 9 × 8) / (3 × 2 × 1) = 720 / 6 = 120. The 7! in the denominator cancels the bottom seven factors of 10!, leaving only 10 × 9 × 8 in the numerator — which is why most small combinations can be worked out mentally once you see the pattern.
For permutations, nPr = n! / (n-r)! = 10! / 7! = 10 × 9 × 8 = 720. Comparing the two results reveals a handy shortcut: nPr = nCr × r!, so 10P3 = 120 × 6 = 720. The r! factor is exactly the number of ways to order the r chosen items — every committee of 3 can be arranged in 3! = 6 different leadership orders, which is why there are six times as many permutations as combinations.
The calculator applies these same formulas but cancels factorials iteratively rather than multiplying enormous values directly. That lets it handle large inputs cleanly — 52C5 = 2,598,960 — without overflowing a 15-digit floating point result, and it also reports the nPr/nCr ratio so you can see how much ordering power adds to the count.
Related Calculators
Use our Combination Calculator Guide together with related tools such as Area Calculator, Combination Calculator, Derivative Calculator to plan more accurately. Each calculator runs instantly in your browser with step-by-step guidance.
Written by the CalcMaster Pro Editorial Team — financial, health, and DIY tools reviewed for accuracy. All calculators run on standard, widely accepted formulas. Always confirm final numbers with a qualified professional for decisions that require official figures.