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.
With Replacement
220
CR(10, 3)
Without Replacement
120
C(10, 3)
Number of distinct types/categories
Number of items to select
Show C(n,r) without replacement for comparison
Ways to select 3 items from 10 types (items can repeat)
220
CR(10, 3)
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.
Click to calculate:
| 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 |
CR(n, r) = C(n+r-1, r)
= (n+r-1)! / (r! x (n-1)!)
Order: No | Replacement: Yes
C(n, r) = n! / (r! x (n-r)!)
Order: No | Replacement: No
With Replacement
220
CR(10, 3)
Without Replacement
120
C(10, 3)
Quick-start with common scenarios
Test your skills with practice problems
Practice with 2 problems to test your understanding.
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.
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.
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.
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
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With Replacement
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Without Replacement
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