Introduction
A combination is a way of choosing items from a group where the order does not matter. For example, if you pick 3 fruits from a basket of 10, it doesn't matter which fruit you grab first — you still end up with the same group of 3. The combination formula, C(n, r) = n! / (r! × (n − r)!), tells you exactly how many different groups you can make.
This Combination Calculator lets you quickly find the number of possible selections from any set. Enter your total number of objects (n) and how many you want to choose (r), and the tool does the rest. It supports standard combinations, combinations with repetition, permutations (where order does matter), and even multiset permutations. Every calculation comes with a step-by-step solution, a visual illustration, sample selections, and a chart so you can clearly see how the results change as you adjust your values.
How to Use Our Combination Calculator
Enter the total number of objects and how many you want to choose, then adjust the settings to match your problem. The calculator will give you the exact count of possible selections, a step-by-step solution, a visual illustration, and a chart.
n (total number of objects): Type the total number of items in your set. This is the full group you are picking from. It must be a whole number from 0 to 999,999.
r (number chosen): Type how many items you want to pick from the set. This must be a whole number. For combinations and permutations without repetition, r cannot be larger than n.
Does order matter?: Turn this toggle on if the order of your chosen items matters. When off, the calculator finds combinations (where {1, 2, 3} is the same as {3, 2, 1}). When on, it finds permutations (where order counts and those would be two different results).
Is repetition allowed?: Turn this toggle on if you are allowed to pick the same item more than once. Leave it off if each item can only be chosen one time.
Items have limited repetitions: Turn this toggle on for multiset permutations, where each item can appear a set number of times. When active, the calculator will show an input field for each item so you can type how many copies of that item exist. This is useful for problems like counting the number of ways to arrange the letters in a word that has repeated letters.
What Are Combinations?
A combination is a way of picking items from a group where the order does not matter. For example, if you choose 3 fruits from a basket of 10, it doesn't matter whether you pick the apple first or last — you still end up with the same group of 3 fruits. This is different from a permutation, where the order does matter.
The Combination Formula
To find the number of combinations, we use this formula:
C(n, r) = n! / (r! × (n − r)!)
Here, n is the total number of items you can choose from, and r is the number of items you are choosing. The exclamation mark (!) means factorial, which is the result of multiplying a number by every whole number below it down to 1. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.
Combinations vs. Permutations
The key difference between combinations and permutations is whether order matters. Choosing players A, B, and C for a team is the same combination no matter what order you list them in. But if you're assigning them to first place, second place, and third place, the order matters — that's a permutation. Permutations always produce a larger number than combinations because each unique group can be arranged in multiple ways. You can explore ordered arrangements in more detail with our Permutation Calculator.
Repetition: Allowed or Not?
Sometimes you're allowed to pick the same item more than once. This is called combinations with repetition. For example, if you're ordering 3 scoops of ice cream and you can repeat flavors, the formula changes to:
C'(n, r) = (n + r − 1)! / (r! × (n − 1)!)
When repetition is not allowed, each item can only be chosen once, and you use the standard combination formula shown above.
Multiset Permutations
A multiset permutation applies when you have a collection of items where some are identical. For instance, if you want to find how many different ways you can arrange the letters in the word "APPLE," you have repeated letters (P appears twice). The formula is:
n! / (n₁! × n₂! × … × nₖ!)
Here, n is the total number of items and n₁, n₂, and so on are the counts of each repeated item.
Real-World Examples
- Lottery numbers: Picking 6 numbers out of 49 is a combination without repetition. C(49, 6) = 13,983,816 possible tickets.
- Pizza toppings: Choosing 3 toppings from a menu of 12 where order doesn't matter gives you C(12, 3) = 220 possible pizzas.
- Team selection: A coach selecting 5 starters from 15 players uses C(15, 5) = 3,003 possible lineups.
- Passwords: If a 4-digit PIN uses digits 0–9 and repetition is allowed with order mattering, that's a permutation with repetition: 10⁴ = 10,000 possibilities.
Connections to Other Math Concepts
Combinatorics is closely related to many areas of mathematics and statistics. When analyzing the results of combination problems, you may need to compute summary statistics such as the standard deviation or use the Z score to understand probability distributions. If you're working with data sets generated by combinatorial analysis, the IQR Calculator can help you identify spread and outliers. Additionally, understanding percentages and percent change is often useful when comparing the likelihood of different combinatorial outcomes, and tools like the Percent Error Calculator can help when verifying experimental results against theoretical combinatorial predictions.
How to Use This Calculator
Enter your value for n (the total number of objects) and r (the number you are choosing). Then use the toggle switches to set whether order matters, whether repetition is allowed, or whether you're working with a multiset. Press Calculate to see the result, a step-by-step breakdown of the math, a visual illustration, sample selections, and a chart showing how the result changes as r varies.