Permutation & Combination Calculator: Complete Guide
Our free online permutation and combination calculator helps you quickly calculate nPr (permutations) and nCr (combinations) with or without repetition. Whether you're studying probability, statistics, or working on combinatorics problems, this tool provides instant results with detailed formulas and explanations.
What are Permutations and Combinations?
Permutations and combinations are fundamental concepts in combinatorics that help us count the number of ways to arrange or select items from a set. The key difference is whether order matters (permutation) or doesn't matter (combination).
Permutations (nPr) - Order Matters
A permutation is an arrangement of objects in a specific order. When calculating permutations, ABC is different from BAC because the order is different. Permutations are used when the sequence or arrangement matters.
Permutation Without Repetition
Formula: P(n,r) = n! / (n-r)!
This calculates the number of ways to arrange r items selected from n items, where each item can only be used once. For example, arranging 3 books from a shelf of 5 books where each book can only be selected once.
Permutation With Repetition
Formula: P(n,r) = n^r
This calculates arrangements where items can be repeated. For example, a 4-digit PIN where each digit (0-9) can be used multiple times.
Combinations (nCr) - Order Doesn't Matter
A combination is a selection of objects where order doesn't matter. ABC is the same as BAC in combinations. Combinations are used when you only care about which items are selected, not their arrangement.
Combination Without Repetition
Formula: C(n,r) = n! / (r! × (n-r)!)
This calculates the number of ways to choose r items from n items where each item can only be selected once. For example, selecting 5 cards from a deck of 52 cards for a poker hand.
Combination With Repetition
Formula: C(n,r) = (n+r-1)! / (r! × (n-1)!)
This calculates selections where items can be repeated. For example, choosing 3 scoops of ice cream from 5 flavors where you can pick the same flavor multiple times.
Real-World Applications
- Lottery & Gambling: Calculate odds of winning by determining total possible combinations
- Password Security: Determine the number of possible passwords or PINs for security analysis
- Team Selection: Calculate ways to form teams or committees from a larger group
- Scheduling: Determine possible arrangements for events, meetings, or tasks
- Genetics: Calculate possible genetic combinations in inheritance patterns
- Card Games: Determine probabilities in poker, bridge, and other card games
- Quality Control: Sample selection in statistical quality control
- Cryptography: Analyze encryption key possibilities
Common Examples Explained
Example 1: Race Podium (Permutation)
In a race with 10 runners, how many ways can the top 3 positions be filled? Use P(10,3) = 10!/(10-3)! = 720 ways. Order matters because 1st place is different from 2nd place.
Example 2: Lottery Selection (Combination)
In a lottery where you pick 6 numbers from 49, how many possible combinations exist? Use C(49,6) = 49!/(6!×43!) = 13,983,816 combinations. Order doesn't matter.
Example 3: Password Creation (Permutation with Repetition)
How many 4-digit PINs are possible using digits 0-9? Use P(10,4) = 10^4 = 10,000 possible PINs. Digits can repeat.
Example 4: Ice Cream Selection (Combination with Repetition)
Choosing 3 scoops from 5 flavors where you can repeat flavors. Use C(5+3-1,3) = C(7,3) = 35 combinations.
Key Formulas Summary
- Factorial: n! = n × (n-1) × (n-2) × ... × 2 × 1
- Permutation (no repetition): P(n,r) = n!/(n-r)!
- Permutation (with repetition): P(n,r) = n^r
- Combination (no repetition): C(n,r) = n!/(r!×(n-r)!)
- Combination (with repetition): C(n,r) = (n+r-1)!/(r!×(n-1)!)
Tips for Using the Calculator
- Understand the problem: Determine if order matters (permutation) or not (combination)
- Check for repetition: Can items be selected multiple times?
- Verify n and r: Ensure n (total items) ≥ r (items to choose) for non-repetition cases
- Large numbers: Results can grow very large; the calculator handles up to n=170
- Use examples: Click preset examples to see how different scenarios work
Frequently Asked Questions
What's the difference between permutation and combination?
Permutations count arrangements where order matters (ABC ≠ BAC). Combinations count selections where order doesn't matter (ABC = BAC). Use permutations for rankings, sequences, or arrangements. Use combinations for selections, groups, or teams.
When should I use repetition formulas?
Use repetition formulas when items can be selected or used multiple times. For example, a PIN code allows the same digit multiple times, or choosing ice cream scoops where you can pick the same flavor twice.
Why is P(n,r) always ≥ C(n,r)?
Permutations are always greater than or equal to combinations because permutations count all different arrangements of the same selection. For example, selecting ABC can be arranged in 6 ways (ABC, ACB, BAC, BCA, CAB, CBA) for permutations, but counts as just 1 combination.
What does n! (factorial) mean?
Factorial (n!) means multiplying all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials grow very quickly and are fundamental to permutation and combination formulas.
Can r be greater than n?
For permutations and combinations without repetition, r cannot exceed n (you can't select more items than available). However, with repetition allowed, r can be greater than n since items can be reused.
Privacy & Security
All calculations are performed locally in your browser using JavaScript. No data is sent to any server. Your calculations are completely private and secure.