Combinations Calculator
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Understanding Combinations
What are Combinations?
Combinations represent the number of ways to select items from a set where order doesn't matter. The basic formulas are:
Standard Combination: C(n,r) = n! / (r!(n-r)!)
With Repetition: C(n+r-1,r) = (n+r-1)! / (r!(n-1)!)
Binomial Coefficient: (n r) = n! / (r!(n-r)!)
Types of Combinations
- Standard: Order doesn't matter, no repetition
- With Repetition: Items can be repeated
- Multiset: Some items are identical
- Pascal's Triangle: Visual representation
Important Properties
Symmetry
C(n,r) = C(n,n-r)
Pascal's Rule
C(n,r) = C(n-1,r-1) + C(n-1,r)
Constraints
0 ≤ r ≤ n
Special Cases
C(n,0) = C(n,n) = 1
Common Examples
| Type | Example | Formula | Result |
|---|---|---|---|
| Standard | C(5,2) | 5!/(2!(5-2)!) | 10 |
| Repetition | C(3+2-1,2) | 4!/(2!2!) | 6 |
| Multiset | {1,1,2,3} | 4!/(2!1!1!) | 12 |
| Special | C(n,0) | n!/(0!n!) | 1 |
Applications
Probability
Calculating sample spaces
Binomial Expansion
(x + y)ⁿ coefficients
Set Theory
Subset selection
Real-World Applications
Statistics
Sample selection and probability
Biology
Genetic combinations and patterns
Computer Science
Algorithm analysis and optimization