Combinations with Repetition Calculator
Calculate combinations with repetition using the formula C'(n,r) = (r+n-1)! / (r!(n-1)!)
Calculate Combinations with Repetition
Total distinct objects available for selection
Number of objects to choose (repetition allowed)
Combination with Repetition Results
Formula: C'(n,r) = (r+n-1)! / (r!(n-1)!)
Where: n = 0, r = 0
Example Calculation
5-Digit Combinations Example
Problem: How many combinations of 5 digits with repetition from 0-9?
Total objects (n): 10 (digits 0-9)
Sample size (r): 5
Solution
C'(10,5) = (5+10-1)! / (5!(10-1)!)
C'(10,5) = 14! / (5! × 9!)
C'(10,5) = 87,178,291,200 / (120 × 362,880)
C'(10,5) = 87,178,291,200 / 43,545,600
C'(10,5) = 2,002
Key Concepts
With vs Without Repetition
- With repetition: Elements can repeat
- Without repetition: Each element used once
- Example: {1,1,2} is valid with repetition
Multiset Coefficient
- Also called "stars and bars" method
- Counts multisets (sets with repetition)
- Formula: ((n+r-1) choose r)
Formula Comparison
C'(n,r) = (r+n-1)! / (r!(n-1)!)
C(n,r) = n! / (r!(n-r)!)
Quick Tips
Elements can be repeated in combinations
Order doesn't matter (like regular combinations)
Always gives more results than without repetition
Used in probability and statistical sampling
Understanding Combinations with Repetition
What are Combinations with Repetition?
Combinations with repetition (also called multiset combinations) are selections of objects from a set where:
- •The order of selection doesn't matter
- •Each object can be selected multiple times
- •Objects are treated as distinct types
Real-World Applications
- •Ice cream flavor combinations (multiple scoops)
- •Pizza topping combinations (multiple of same topping)
- •Card draws with replacement
- •Distribution of identical items into different groups
The Multiset Formula
C'(n,r) = (r+n-1)! / (r!(n-1)!)
- C'(n,r): Number of combinations with repetition
- n: Total number of distinct objects
- r: Number of selections to make
- ! Factorial notation
Stars and Bars Method
This formula is equivalent to choosing r items from n types, which is the same as placing r identical balls into n distinct urns.
Alternative Form: C'(n,r) = C(r+n-1, r) where C is the regular combination formula. This shows the connection to binomial coefficients.
Memory Tip: With repetition always gives MORE combinations than without, since you have more choices available when elements can be reused.