Permutation and Combination Calculator
Calculate permutations (nPr) and combinations (nCr) with step-by-step solutions and examples
Calculate Permutations and Combinations
Total number of objects available (maximum 170 for accurate calculation)
Number of objects to select (must be ≤ n)
Results
Permutations (nPr)
Formula: P(5,3) = 5! / (5-3)! = 5! / 2!
Order matters - different arrangements of the same items are counted separately
Combinations (nCr)
Formula: C(5,3) = 5! / (3! × (5-3)!) = 5! / (3! × 2!)
Order doesn't matter - different arrangements of the same items are counted as one
Key Formulas
• Permutation: nPr = n! / (n-r)!
• Combination: nCr = n! / (r!(n-r)!)
• Factorial: n! = n × (n-1) × (n-2) × ... × 1
• Relationship: nCr = nPr / r!
Step-by-Step Calculation
P(5,3) = 5! / (5-3)! = 5! / 2!
P(5,3) = 120 / 2 = 60
C(5,3) = 5! / (3! × (5-3)!) = 5! / (3! × 2!)
C(5,3) = 120 / (6 × 2) = 10
All Possible Results
All 10 Combinations:
All 60 Permutations:
Example Calculations
Permutation Example
Problem: How many ways can you arrange 3 books from 5 books?
Solution: P(5,3) = 5!/(5-3)! = 120/2 = 60
Answer: 60 different arrangements
Real-world Application
• Race finishing positions
• Password arrangements
• Seating arrangements
• Tournament brackets
Combination Example
Problem: How many ways can you choose 3 books from 5 books?
Solution: C(5,3) = 5!/(3!×2!) = 120/(6×2) = 10
Answer: 10 different selections
Real-world Application
• Lottery number selection
• Team member selection
• Menu combinations
• Study group formation
Quick Reference
Permutation (nPr)
Formula: n! / (n-r)!
When: Order matters
Example: ABC ≠ BAC
Combination (nCr)
Formula: n! / (r!(n-r)!)
When: Order doesn't matter
Example: ABC = BAC
Key Differences
Permutation
Order is important
Arrangements matter
Combination
Order doesn't matter
Selections only
Factorial
n! = n × (n-1) × ... × 1
Building block for both
Calculator Tips
n must be ≥ r for valid results
Both n and r must be non-negative integers
Large numbers may show scientific notation
nCr is always ≤ nPr for the same n,r
Understanding Permutations and Combinations
What are Permutations?
Permutations are arrangements of objects where the order matters. When calculating permutations, different sequences of the same objects are considered different arrangements.
Permutation Formula
P(n,r) = n! / (n-r)!
- n: Total number of objects
- r: Number of objects to arrange
- P(n,r): Number of permutations
What are Combinations?
Combinations are selections of objects where the order doesn't matter. When calculating combinations, different arrangements of the same objects are considered the same selection.
Combination Formula
C(n,r) = n! / (r!(n-r)!)
- n: Total number of objects
- r: Number of objects to select
- C(n,r): Number of combinations
When to Use Each?
Use Permutations When:
- • Order of selection matters
- • Arranging objects in a sequence
- • Creating passwords or codes
- • Race finishing positions
- • Seating arrangements
Use Combinations When:
- • Order of selection doesn't matter
- • Choosing team members
- • Selecting lottery numbers
- • Picking committee members
- • Choosing menu items
Important Properties
nC0 = 1 (choosing nothing)
nCn = 1 (choosing everything)
nC1 = n (choosing one item)
nP0 = 1 (arranging nothing)
nPn = n! (arranging everything)
nCr = nC(n-r) (symmetry property)