Combinations with Replacement Calculator
Calculate combinations with replacement CR(n,r) - selections where items can be repeated and order does not matter. Get step-by-step solutions and practical examples.
CR(n,r) = C(n+r-1, r) = (n+r-1)! / (r! x (n-1)!)Results
With Replacement
220
CR(10, 3)
Without Replacement
120
C(10, 3)
Input Values
Number of distinct types/categories
Number of items to select
Show C(n,r) without replacement for comparison
Results
With Replacement
Ways to select 3 items from 10 types (items can repeat)
220
CR(10, 3)
Without Replacement
Ways to select 3 items from 10 (no repeats)
120
C(10, 3)
Ratio: CR(10,3) / C(10,3) = 1.83x more combinations when replacement is allowed.
Step-by-Step Calculation
Practical Examples
Click to calculate:
Combinations Comparison
| Type | Order | Replacement | Formula | Result |
|---|---|---|---|---|
| CR (this) | No | Yes | C(n+r-1, r) | 220 |
| C (standard) | No | No | n!/(r!(n-r)!) | 120 |
| PR (perm w/ rep) | Yes | Yes | n^r | 1,000 |
| P (standard) | Yes | No | n!/(n-r)! | 720 |
Formulas Reference
Combinations with Replacement
CR(n, r) = C(n+r-1, r)
= (n+r-1)! / (r! x (n-1)!)
Order: No | Replacement: Yes
Standard Combinations
C(n, r) = n! / (r! x (n-r)!)
Order: No | Replacement: No
Results
With Replacement
220
CR(10, 3)
Without Replacement
120
C(10, 3)
?How Do You Calculate Combinations with Replacement?
Combinations with replacement (multiset) counts selections where items CAN be repeated and order does NOT matter. Formula: CR(n,r) = C(n+r-1, r) = (n+r-1)! / (r! x (n-1)!). Example: Choosing 3 ice cream scoops from 5 flavors (can repeat) = CR(5,3) = C(7,3) = 35 ways.
What are Combinations with Replacement?
Combinations with replacement, also called multiset combinations, count the number of ways to select r items from n types where repetition is allowed and order does not matter. The formula CR(n,r) = C(n+r-1, r) uses the 'stars and bars' technique. This is useful for problems like choosing ice cream scoops, distributing identical items, or selecting with replacement from a set.
Key Facts
- Combinations with replacement allow selecting the same item multiple times
- Formula: CR(n,r) = C(n+r-1, r) = (n+r-1)! / (r! x (n-1)!)
- Also known as multiset combinations or combinations with repetition
- Order does NOT matter: choosing AAB is the same as ABA or BAA
- Uses the "Stars and Bars" counting technique in combinatorics
- CR(n,r) >= C(n,r) since replacement allows more possibilities
- Example: Ice cream scoops, dice outcomes (unordered), card draws with replacement
- Related to distributing r identical items into n distinct bins
Quick Answer
Combinations with replacement (multiset) counts selections where items CAN be repeated and order does NOT matter. Formula: CR(n,r) = C(n+r-1, r) = (n+r-1)! / (r! x (n-1)!). Example: Choosing 3 ice cream scoops from 5 flavors (can repeat) = CR(5,3) = C(7,3) = 35 ways.
Frequently Asked Questions
Combinations with replacement (multiset combinations) count selections where items CAN be repeated and order does NOT matter. For example, choosing 3 ice cream scoops from 5 flavors where you can pick the same flavor multiple times. Formula: CR(n,r) = C(n+r-1, r).
The formula is CR(n,r) = C(n+r-1, r) = (n+r-1)! / (r! x (n-1)!). This is derived from the "stars and bars" counting technique, where we count ways to distribute r identical items into n distinct categories.
WITHOUT replacement: each item can only be selected once (standard combinations). WITH replacement: items can be selected multiple times (multiset). CR(n,r) is always >= C(n,r). Example: C(5,3)=10 but CR(5,3)=35.
Stars and bars is a counting method to find ways to distribute r identical items (stars) into n distinct bins separated by n-1 dividers (bars). The total arrangements of (r stars + n-1 bars) choosing positions for r stars gives C(n+r-1, r).
Use when: (1) Order does not matter (unordered selection), (2) Items CAN be repeated. Examples: ice cream scoops, dice sum distributions (unordered), selecting fruits from unlimited supply, multisets in mathematics.
Last updated: 2025-01-15
Results
With Replacement
220
CR(10, 3)
Without Replacement
120
C(10, 3)