Permutation & Combination Calculator
Instantly calculate nPr and nCr with full step-by-step factorial working. Perfect for students, teachers, and anyone working with probability and statistics.
Calculate nPr and nCr
📊 Results for n = , r =
Permutation (nPr)
Combination (nCr)
📝 Step-by-Step Working
How to Use the Calculator
- Enter the total number of items in the set in the n field. This must be a non-negative integer between 0 and 170.
- Enter the number of items to select or arrange in the r field. r must be less than or equal to n.
- Click the Calculate button to instantly see both the permutation nPr and combination nCr results with step-by-step factorial working.
- Review the displayed nPr and nCr values along with the expanded factorial formulas to understand how each result was computed.
Key Features
Instant Dual Results
Calculates both nPr and nCr simultaneously with a single click, saving time for students and professionals.
Step-by-Step Breakdown
Shows complete factorial expansion so you understand the working, not just the final answer.
Large Number Support
Handles n values up to 170 and displays very large results clearly, including scientific notation where needed.
Smart Validation
Catches invalid inputs like r greater than n, negative values, and decimal entries before calculation.
Works on All Devices
Fully responsive layout works perfectly on mobiles, tablets, and desktops without any installation.
100% Client-Side
All calculations happen in your browser. No data is sent to any server, keeping your inputs private.
Formulas and How It Works
Permutations and combinations are foundational concepts in combinatorics. The key difference is whether the order of selection matters.
Permutation Formula (nPr)
Permutation counts the number of ways to arrange r items chosen from n items where order matters.
Where:
n = total number of items
r = number of items selected
! = factorial (product of all positive integers up to that number)
Combination Formula (nCr)
Combination counts the number of ways to choose r items from n items where order does not matter.
Also written as C(n,r) or "n choose r"
Relationship: nPr = nCr × r!
Factorial Definition
Example: 5! = 5 × 4 × 3 × 2 × 1 = 120
Special case: 0! = 1 (by definition)
Practical Examples
🇮🇳 Priya — Delhi, India (Board Exam Preparation)
Priya is a Class 11 student preparing for her CBSE board exams. Her math textbook asks: "In how many ways can a committee of 3 be selected from 8 students?" and "In how many ways can a president, VP, and secretary be chosen from 8 students?"
She enters n = 8, r = 3 into the calculator.
nPr = 8! / 5! = 336 (arrangements — order matters for the officer roles). nCr = 8! / (3! × 5!) = 56 (selections — order doesn't matter for the committee).
🇮🇳 Rakesh — Bengaluru, India (Software Developer)
Rakesh is designing a 4-digit PIN system for an app. He needs to know how many unique 4-digit PINs can be formed using 10 digits (0–9) without repeating any digit.
He enters n = 10, r = 4 into the calculator.
nPr = 10! / 6! = 5,040 unique PINs. Since order matters (1234 ≠ 4321), permutation is the correct formula here.
🇮🇳 Ananya — Mumbai, India (Data Science Student)
Ananya is studying for her statistics exam and needs to calculate the probability of winning a lottery where 6 numbers are drawn from 49 without replacement, and order doesn't matter.
She enters n = 49, r = 6 into the calculator.
nCr = 49! / (6! × 43!) = 13,983,816 possible combinations. The probability of winning is 1 in 13,983,816.
🇺🇸 James — New York, USA (Sports Analyst)
James is analyzing basketball team selections. A coach needs to choose 5 starting players from a squad of 12. He wants to know how many possible starting lineups exist.
He enters n = 12, r = 5 into the calculator.
nCr = 12! / (5! × 7!) = 792 possible starting lineups. Since the starting five roles aren't ranked, combination is correct here.
What Is a Permutation & Combination Calculator?
A Permutation and Combination Calculator is a mathematical tool that computes the number of possible arrangements (permutations) and selections (combinations) from a given set of items. These calculations are central to probability theory, statistics, computer science, and everyday decision-making involving choices and arrangements.
Permutations (nPr) answer the question: "How many ways can I arrange r things from a set of n, where the sequence matters?" Combinations (nCr) answer: "How many ways can I choose r things from n, where the sequence doesn't matter?" The same n and r values always produce a permutation result that is equal to or larger than the combination result, since permutations account for all possible orderings of each selected group.
This tool is widely used by school and college students for probability chapters, by data scientists for statistical sampling calculations, by software developers for algorithm design, and by puzzle and game designers computing solution spaces. Instead of computing large factorials manually, this calculator delivers instant, verified results with complete working shown.
Permutation & Combination in Multiple Languages
📖 Want to understand permutations and combinations with more examples?
Read Our Complete Guide to Permutations & Combinations →Frequently Asked Questions
Is this Permutation & Combination Calculator free to use?
Yes, this calculator is completely free to use with no registration, no signup, and no hidden charges. You can calculate unlimited permutations and combinations anytime.
What is the difference between permutation and combination?
Permutation (nPr) counts arrangements where order matters — selecting 3 from 5 gives different results based on sequence. Combination (nCr) counts selections where order does not matter — choosing 3 from 5 gives the same group regardless of arrangement order.
What is the formula for permutation nPr?
The permutation formula is nPr = n! / (n - r)! where n is the total number of items and r is the number of items selected. The exclamation mark denotes factorial.
What is the formula for combination nCr?
The combination formula is nCr = n! / (r! × (n - r)!) where n is the total number of items and r is the number of items chosen. The result is always less than or equal to nPr.
What does factorial mean?
Factorial of a number n (written as n!) means the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition, 0! = 1.
Can r be greater than n?
No. r cannot be greater than n. If r exceeds n, there are not enough items to select or arrange, and the result is mathematically undefined. The calculator will show a validation error in this case.
What is the maximum value of n supported?
The calculator supports n values up to 170. Beyond this, factorials exceed JavaScript's maximum safe number representation. For very large n values, results are shown in scientific notation.
How is this calculator useful in real life?
Permutations are used in password generation, race rankings, and scheduling. Combinations are used in lottery odds, team selection, card games, and statistical sampling. Both are core concepts in probability and statistics.
What happens when r equals 0?
When r equals 0, nPr = 1 and nCr = 1. This is because there is exactly one way to arrange zero items (do nothing) and exactly one way to choose zero items from a set.
What is the relationship between nPr and nCr?
nPr = nCr × r! because a permutation is a combination where the selected items are then arranged in all possible orders. So nCr multiplied by the number of ways to arrange r items (r!) gives nPr.
Recommended Hosting
Hostinger
If you are building a website for your tools, blog, or store, reliable hosting matters for speed and uptime. Hostinger is a popular option used worldwide.
Visit Hostinger →Disclosure: This is a sponsored link.
Contact Us
💬 Chat with us on WhatsApp
WhatsApp: +91 92580 36351📧 Send us an email
contact@storedropship.in