Permutation & Combination Calculator

Permutation & Combination Calculator Permutations Combinations Advanced Examples ...

Permutation & Combination Calculator

Permutations (Order Matters)

Total Items (n)

Items to Choose (r)

Permutation Type

Without Repetition (P(n,r)) With Repetition (nʳ)

💡 Formula:
P(n,r) = n! / (n-r)!

Combinations (Order Doesn't Matter)

Total Items (n)

Items to Choose (r)

Combination Type

Without Repetition (C(n,r)) With Repetition (C(n+r-1,r))

💡 Formula:
C(n,r) = n! / (r!(n-r)!)

Advanced Calculations

Calculation Type

Factorial (n!) Partial Permutations Multiset Permutations Circular Permutations

Number (n)

Items to Choose (r)

Repeated Elements

💡 Formulas:
n! = n × (n-1) × ... × 1

Common Examples

🔐 Password Creation
4-digit password using digits 0-9

👥 Committee Selection
Choose 3 people from 10 for a committee

🏆 Race Finishing Order
Top 3 finishers from 8 runners

🃏 Card Hands
5-card poker hand from 52 cards

📚 Book Arrangement
Arrange 5 different books on a shelf

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Click "Print or Save as PDF" above → Choose "Save as PDF" as your printer → Click "Save".

Results

Total Items (n): 5

Items to Choose (r): 3

Permutation Type: Without Repetition

Result: 60

Formula Used: P(5,3) = 5! / (5-3)! = 120 / 2 = 60

Explanation: 60 different ways to arrange 3 items from 5 when order matters

Permutation Example

Items: A, B, C, D, E

Choosing 3 with order mattering:

ABC ACB BAC BCA CAB CBA ...

Total: 60 arrangements

Total Items (n): 5

Items to Choose (r): 3

Combination Type: Without Repetition

Result: 10

Formula Used: C(5,3) = 5! / (3!(5-3)!) = 120 / (6 × 2) = 10

Explanation: 10 different ways to choose 3 items from 5 when order doesn't matter

Combination Example

Items: A, B, C, D, E

Choosing 3 without order mattering:

ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE

Total: 10 combinations

Calculation Type: Factorial

Input: 5

Result: 120

Formula Used: 5! = 5 × 4 × 3 × 2 × 1 = 120

Explanation: 120 ways to arrange 5 distinct items

Example: Password Creation

Problem: 4-digit password using digits 0-9

Type: Permutation with Repetition

Result: 10,000

Formula: 10⁴ = 10,000

Explanation: Each digit has 10 choices (0-9), so 10×10×10×10 = 10,000 possible passwords

Understanding Permutations and Combinations

Permutations and combinations are fundamental concepts in combinatorics, the branch of mathematics dealing with counting, arrangement, and selection of objects. Understanding the difference between them is crucial for solving probability problems, statistical analysis, and various real-world applications.

Permutations deal with arrangements where order matters. For example, if you're selecting a president, vice-president, and secretary from a group, the order of selection is important - ABC is different from BAC. The formula for permutations without repetition is P(n,r) = n! / (n-r)!.

Combinations deal with selections where order doesn't matter. Using the same example, if you're just selecting a 3-person committee, then ABC is the same as BAC - the order doesn't matter. The formula for combinations without repetition is C(n,r) = n! / (r!(n-r)!).

These concepts are essential in fields like computer science (algorithm design), cryptography (password security), genetics (gene combinations), and game theory (probability calculations).

Frequently Asked Questions

Q: How do I know whether to use permutation or combination?
A: Ask yourself: "Does the order matter?" If yes, use permutation. If no, use combination. For example, lottery numbers (order doesn't matter) use combinations, while race finishing positions (order matters) use permutations.

Q: What's the difference between with and without repetition?
A: Without repetition means each item can only be used once (like drawing cards from a deck). With repetition means items can be reused (like digits in a password). Permutations with repetition: nʳ. Combinations with repetition: C(n+r-1, r).

Q: What is factorial and how is it calculated?
A: Factorial (n!) is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition, 0! = 1. Factorials grow very quickly - 10! = 3,628,800.

Q: Can I have more items to choose than total items (r > n)?
A: For permutations and combinations without repetition, if r > n, the result is 0 (impossible). For combinations with repetition, it's still possible even when r > n because you can repeat items.

Q: What are circular permutations?
A: Circular permutations arrange items in a circle where rotations are considered the same. The formula is (n-1)! instead of n! because rotating the entire arrangement doesn't create a new permutation.

Q: How do repeated elements affect permutations?
A: When some items are identical, you divide by the factorial of the counts of each repeated item. For example, arrangements of "AAB" would be 3! / 2! = 3, since the two A's are indistinguishable.

Q: What's the relationship between permutations and combinations?
A: The relationship is P(n,r) = C(n,r) × r!. This makes sense because each combination of r items can be arranged in r! different ways (permutations), so total permutations equal combinations times arrangements per combination.