Combination Calculator
Calculate combinations (nCr) and permutations (nPr) with detailed mathematical explanations
Calculate Combinations & Permutations
Total number of distinct objects in the set
Number of objects to choose from the set
Number of Selections
Mathematical Formulas
Combinations: C(n,r) = n! / (r! × (n-r)!) = 10! / (3! × 7!)
Example: Colorful Balls
Problem Setup
Question: You have 12 different colored balls. How many ways can you choose 5 balls?
Given: n = 12, r = 5
Solution
C(12,5) = 12! / (5! × 7!)
C(12,5) = 95,040 / (120 × 5,040)
C(12,5) = 792 ways
Key Differences
Combinations
Order doesn't matter
ABC = ACB = BAC
Permutations
Order matters
ABC ≠ ACB ≠ BAC
With Repetition
Elements can be chosen multiple times
AAA, ABB allowed
Mathematical Tips
C(n,r) = C(n,n-r) - combinations are symmetric
P(n,r) = C(n,r) × r! - permutations from combinations
C(n,0) = 1 and C(n,n) = 1 always
Permutations ≥ Combinations (when r > 1)
Understanding Combinations and Permutations
What Are Combinations?
A combination is a selection of items from a collection where the order of selection does not matter. When you choose r elements from n distinct objects, you're calculating "n choose r" or C(n,r). For example, choosing 3 people from a group of 10 for a committee.
When to Use Combinations
- •Selecting team members from a group
- •Choosing items from a menu
- •Lottery number selection
- •Portfolio selection in finance
What Are Permutations?
A permutation is an arrangement of items from a collection where the order of arrangement matters. When you arrange r elements from n distinct objects, you're calculating P(n,r). For example, arranging 3 people in a line from a group of 10.
When to Use Permutations
- •Arranging people in a line or seating
- •Creating passwords or codes
- •Race finishing positions
- •Sequential decision making
Mathematical Formulas
Without Repetition
Combinations: C(n,r) = n! / (r! × (n-r)!)
Permutations: P(n,r) = n! / (n-r)!
Relationship: P(n,r) = C(n,r) × r!
With Repetition
Combinations: C(n+r-1,r) = (n+r-1)! / (r! × (n-1)!)
Permutations: P(n,r) = n^r
Note: Elements can be chosen multiple times
Key Insight
The fundamental difference is whether order matters. If you're selecting a committee, order doesn't matter (combination). If you're assigning positions like president, vice-president, and secretary, order matters (permutation).