nCr Calculator

Category: Statistics

- May 15, 2025

| |

Calculate combinations (nCr), permutations (nPr), factorials, and other related values. This calculator helps solve probability, statistics, and combinatorial problems.

Input Values

n (total items):

Number of distinct items in the set

r (items to choose):

Number of items to select from the set

Calculation Options:

Allow repetition (replacement)

Select if items can be used more than once

Formulas and Definitions

Combinations (without repetition):

C(n,r) = n!r!(n-r)!

Number of ways to choose r items from a set of n items, where order doesn't matter.

Permutations (without repetition):

P(n,r) = n!(n-r)!

Number of ways to arrange r items from a set of n items, where order matters.

Combinations (with repetition):

C(n+r-1,r) = (n+r-1)!r!(n-1)!

Number of ways to choose r items from a set of n items, where order doesn't matter and repetition is allowed.

Permutations (with repetition):

n^r

Number of ways to arrange r items from a set of n items, where order matters and repetition is allowed.

Factorial:

n! = n × (n-1) × (n-2) × ... × 2 × 1

Product of all positive integers less than or equal to n. By definition, 0! = 1.

Applications

Probability Theory: Calculate probabilities of compound events
Statistics: Binomial probability distributions and sampling
Game Theory: Calculate possible game states and outcomes
Computer Science: Algorithm analysis and data structure optimization
Genetics: Gene combinations and population genetics

Note: This calculator handles values up to 170 for factorials due to JavaScript's numerical limitations. For larger values, approximations may be used.

Combination (nCr) Formula:
C(n, r) = n! / (r! × (n − r)!)

Permutation (nPr) Formula:
P(n, r) = n! / (n − r)!

Combinations with Repetition:
C(n + r − 1, r) = (n + r − 1)! / (r! × (n − 1)!)

Permutations with Repetition:
n^r

What Is the nCr Combination Calculator?

The nCr Combination Calculator is a practical tool for solving problems involving combinations, permutations, and factorials. It is especially useful for students, teachers, and professionals working with probability and statistical analysis. Whether you're calculating how many ways you can choose items from a group or figuring out different possible arrangements, this tool saves time and improves accuracy.

Why Use This Calculator?

This calculator serves as a:

Statistics tool to simplify combinatorial analysis and arrangements
Probability and stats helper for evaluating chances and outcomes
Permutation and combination guide for structured problem-solving
Statistical computation resource when working with factorial-based formulas
Data analysis helper for interpreting choices in surveys, games, or experimental designs

How to Use the Calculator

Enter the total number of items in the n field (e.g., 10).
Enter the number of items to select in the r field (e.g., 4).
If repetition is allowed (items can be chosen more than once), check the "Allow repetition" box.
Click "Calculate" to see the results.
Review the values for combinations, permutations, factorials, and more.
Click "Reset" to clear the form and try new values.

What Does It Calculate?

The calculator instantly provides multiple outputs:

Combinations (nCr): The number of ways to choose items when order doesn't matter
Permutations (nPr): The number of ways to arrange items when order matters
Factorials: Key components for calculating nCr and nPr values
With Repetition: Results when items can be reused in selection
Binomial Coefficient: Also known as nCr, useful in probability distributions
Total Arrangements: The full factorial of n (n!)

Use Cases and Benefits

This combination solver is ideal for:

Probability analysis: Understanding chance and event likelihood
Statistical analysis: Exploring sample selections and data distribution
Descriptive statistics: Supporting calculations involving the mean and median
Combinatorial scenarios: Exam questions, Game setups, or logic puzzles
Educational support: Great for learning how permutations and combinations work

Frequently Asked Questions

Q: What is the difference between a combination and a permutation?

A: A combination focuses on selection where order doesn't matter (e.g., picking 3 fruits from a bowl), while a permutation focuses on arrangement where order matters (e.g., assigning 3 runners to 3 positions).

Q: When should I check the "Allow repetition" box?

A: If items can be chosen more than once, such as drawing marbles with replacement or forming digit codes, enable repetition.

Q: What if I enter values too large?

A: The calculator supports inputs up to 170 due to JavaScript limits. Beyond that, the result will show as ∞ or use scientific notation.

Q: Can this tool help with data Science or statistics homework?

A: Yes. It’s a helpful Statistics Calculator that can assist in tasks like analyzing data sets, calculating probability distributions, or running statistical computations.

Q: How is this different from a basic calculator?

A: Unlike a standard calculator, this one is built specifically for permutations and combinations, showing you step-by-step results and factorials, with additional visual tools like Pascal's Triangle.

Conclusion

Whether you're studying probability, solving a combinatorial puzzle, or performing statistical computations, the nCr Combination Calculator gives you a quick and clear answer. It's a powerful yet easy-to-use combination solver and permutation formula tool. Try it now and make your calculations faster, simpler, and more insightful.