What's the difference between permutation and combination?

The key difference is order. Permutations are arrangements where order matters (e.g., ABC and CBA are different). Combinations are selections where order does not matter (e.g., {A,B,C} and {C,B,A} are the same group).

When should I use permutations vs combinations?

Use permutations for ordered lists, like arranging books on a shelf, race results, or assigning specific roles. Use combinations for unordered groups, like choosing a committee, picking lottery numbers, or selecting toppings for a pizza.

How do I handle permutations with repetition?

It depends. If you are choosing 'r' items from 'n' with repetition allowed, use n^r (our 'Variation' option). If arranging a set with identical items (like letters in 'APPLE'), use the multiset formula: n! / (n1! n2! ...), where n1 is the count of the first repeated item, etc.

What is a circular permutation?

A circular permutation is an arrangement of items in a circle. Since there's no start or end point, rotations are considered the same arrangement. The formula is (n - 1)!, where 'n' is the number of items. For example, arranging 4 people around a table is (4-1)! = 3! = 6 ways.

What does 0! (zero factorial) equal and why?

0! is defined to be 1. This is a mathematical convention that allows formulas like the permutation formula P(n,n) = n! / (n-n)! to work correctly. It represents one way to arrange nothing (the arrangement of an empty set).

Can 'n' and 'r' be the same in permutations?

Yes. When n = r, you are calculating the number of ways to arrange all items in a set. The formula P(n,n) simplifies to n! / (n-n)! = n! / 0! = n!, which is the total number of unique ways to order the entire set.

Is there a combination with repetition?

Yes. This is often called a 'multiset coefficient' or solved with the 'stars and bars' method. The formula is C(n+r-1, r). It calculates ways to choose 'r' items from 'n' options where order doesn't matter and repetition is allowed, like picking 3 scoops of ice cream from 5 flavors.

How are permutations used in data science?

In machine learning, permutation tests are used to determine feature importance. By shuffling the values of a single feature and measuring how much the model's accuracy drops, data scientists can gauge how critical that feature is for making correct predictions.

What is the factorial notation (!) in permutations?

Factorial (n!) is the product of all positive integers up to n (e.g., 5! = 5 × 4 × 3 × 2 × 1 = 120). It's a cornerstone of permutation and combination formulas because it represents the total number of ways to arrange n distinct items.

How are these concepts used in probability?

They are fundamental. Probability is often calculated as (Number of Favorable Outcomes) / (Total Number of Possible Outcomes). Permutations and combinations are used to find both the numerator and the denominator in this equation. For example, combinations can tell you the total possible poker hands, and the number of ways to get a specific hand (like a full house).

Combination Calculator

Total Items (n)

Items to Choose (r)

Combination Result (nCr)

120

C(10,3) = 10! / (3! × (10-3)!) = 120

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How to Use the Combination Calculator

Calculate combinations and permutations in three simple steps:

1. Enter Values

Input the total number of items (n) and the number to choose (r) in the input fields.

2. Select Calculation Type

Choose between Combination (nCr) or Permutation (nPr) using the mode buttons.

3. Get Instant Results

View the calculated result along with the formula explanation.

Applications of Combinations & Permutations

Understanding combinatorial calculations is essential in many fields:

Probability & Statistics

Calculate probabilities, odds, and statistical significance in research and data analysis.

Cryptography

Determine key space size and encryption strength in security systems.

Genetics & Biology

Analyze genetic combinations and protein sequences in biological research.

Combination vs Permutation

Understanding the difference between these two fundamental concepts:

Feature	Combination (nCr)	Permutation (nPr)
Order Matters	No	Yes
Formula	n! / [r! × (n-r)!]	n! / (n-r)!
Example	Lottery numbers	Password combinations
Use Case	Selecting committee members	Race finishing orders
Values	nCr ≤ nPr	nPr ≥ nCr

Key Formulas:
Combination (nCr): C(n, r) = n! / (r! × (n – r)!)
Permutation (nPr): P(n, r) = n! / (n – r)!

The Foundation: Understanding Factorials

Both combination and permutation formulas rely on the concept of a factorial. A factorial, denoted by an exclamation mark (!), is the product of all positive integers up to that number.

For a non-negative integer n, the factorial (n!) is defined as:

n! = n × (n-1) × (n-2) × … × 1

For example, 5! is calculated as: 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition, the factorial of zero (0!) is 1. Factorials are fundamental in combinatorics because they represent the total number of ways to arrange a set of distinct objects.

Real-World Examples & Solved Problems

Combinatorics isn’t just theoretical; it’s used to solve practical problems every day. Here are a few detailed examples.

Lottery Odds

In a lottery where you pick 6 numbers from 49, the order doesn’t matter. How many possible combinations are there? Use nCr: C(49, 6) = 13,983,816. Your odds are 1 in 14 million.

Committee Selection

A company needs to form a 4-person committee from 15 employees. Since all roles are equal, order doesn’t matter. The number of possible committees is: C(15, 4) = 1,365.

Race Finishing Order

In a race with 8 contestants, how many different ways can the top 3 (Gold, Silver, Bronze) finish? Here, order is crucial. Use nPr: P(8, 3) = 336. There are 336 possible podium finishes.

Advanced Concepts: Combinations with Repetition

Standard combinations assume each item is distinct and chosen only once. But what if you can choose the same item multiple times? This is known as combinations with repetition (or multisets).

Imagine you’re buying 3 scoops of ice cream from a shop with 5 flavors. You could get three scoops of the same flavor. The formula for this is:

C'(n, r) = C(n + r – 1, r) = (n + r – 1)! / (r! × (n – 1)!)

In our ice cream example (n=5, r=3): C(5 + 3 – 1, 3) = C(7, 3) = 35. There are 35 different combinations of 3 scoops you could choose.

Pascal’s Triangle and Its Link to Combinations

Pascal’s Triangle is a triangular array of numbers where each number is the sum of the two directly above it. It has a beautiful, direct relationship with combinations. The value of nCr can be found at the (r+1)th position in the (n+1)th row of the triangle (starting from 0).

Row 0: 1 (C(0,0))
Row 1: 1 1 (C(1,0), C(1,1))
Row 2: 1 2 1 (C(2,0), C(2,1), C(2,2))
Row 3: 1 3 3 1 (C(3,0), C(3,1), C(3,2), C(3,3))
Row 4: 1 4 6 4 1 (C(4,0), C(4,1), C(4,2), C(4,3), C(4,4))

This pattern is not a coincidence. It’s because the formula to build Pascal’s Triangle (Pascal’s Identity) is algebraically equivalent to a fundamental identity of combinations. This makes it a powerful tool in algebra, especially for binomial expansions like (x+y)ⁿ.

Frequently Asked Questions

What is the difference between combination and permutation?

Combinations (nCr) focus on selecting items where order doesn’t matter, while permutations (nPr) consider arrangements where order is important. For example, choosing 3 fruits from a basket is a combination, but arranging them in a specific order is a permutation.

Can I calculate combinations with very large numbers?

Yes, our calculator uses efficient algorithms to handle large values accurately. For extremely large numbers, we utilize scientific notation to provide precise results without crashing your browser.

What if r is greater than n?

In combinatorial mathematics, if r > n, both nCr and nPr are defined as 0. You can’t choose more items than are available. Our calculator will automatically return 0 in such cases and display an appropriate message.

Does the calculator support decimal or fractional values?

No, combinations and permutations are only defined for non-negative integers. The calculator will automatically round any decimal inputs to the nearest integer and display a warning message.

Can I calculate binomial coefficients with this tool?

Yes, binomial coefficients are equivalent to combinations (nCr). Our combination calculator directly computes binomial coefficients, making it perfect for binomial theorem applications, polynomial expansions, and Pascal’s triangle calculations.

How accurate are the calculations?

Our calculator provides mathematically precise results. For values within standard JavaScript number limits, results are exact. For extremely large numbers, we use scientific notation while maintaining high precision.

Can I use this calculator for probability problems?

Absolutely! Combinations and permutations are fundamental to probability calculations. Our tool is perfect for determining sample spaces, calculating odds, and solving probability problems in statistics, genetics, and gaming theory.

How does this differ from other online calculators?

Our calculator provides real-time computation, detailed formula explanations, and handles larger values than most competitors. We also offer educational content, comparison tables, and privacy-focused local processing—no data leaves your device.

What does n! (factorial) mean?

A factorial (n!) is the product of all positive integers up to n (e.g., 4! = 4 x 3 x 2 x 1 = 24). It’s a key component of combination and permutation formulas. We have a dedicated section explaining it above.

Is this calculator suitable for academic use?

Absolutely! Our calculator is trusted by students and educators worldwide. It provides precise results and detailed formula explanations that help with learning combinatorial mathematics. However, always check your institution’s policy on calculator use for exams and assignments.