Possible Combinations Calculator
Calculate combinations with and without repetition using statistical formulas
Calculate Possible Combinations
The total number of distinct objects available
The number of objects to choose from the total
Each object can only be chosen once (no duplicates)
Example: Combinations of 5 Letters
Problem
Question: How many combinations of 5 letters are possible from the English alphabet?
Given: n = 26 (total letters), r = 5 (letters to choose)
Solution
Without repetition: C(26,5) = 26! / (5!(26-5)!) = 65,780
With repetition: C'(26,5) = (5+26-1)! / (5!(26-1)!) = 142,506
Formula Reference
Without Repetition
C(n,r) = n! / (r!(n-r)!)
Each object can only be chosen once. Order doesn't matter.
With Repetition
C'(n,r) = (r+n-1)! / (r!(n-1)!)
Objects can be chosen multiple times. Order doesn't matter.
Key Concepts
Combinations
Selection where order doesn't matter
Permutations
Arrangement where order matters
Factorial
Product of all positive integers up to n
Quick Tips
Use combinations when order doesn't matter
With repetition allows choosing the same element multiple times
C(n,0) = 1 for any n ≥ 0
C(n,r) = C(n,n-r)
Understanding Combinations
What are Combinations?
Combinations are ways to select items from a collection where the order of selection doesn't matter. This is different from permutations where order is important.
When to Use Combinations
- •Selecting team members from a group
- •Choosing lottery numbers
- •Selecting items from a menu
- •Creating subsets from a set
Combinations vs. Permutations
Example: Choosing 2 letters from (A, B, C)
Combinations (order doesn't matter)
- (A, B)
- (A, C)
- (B, C)
Total: 3
Permutations (order matters)
- AB, BA
- AC, CA
- BC, CB
Total: 6
Repetition Explained
Without repetition: Each element can only be chosen once (like drawing cards without replacement)
With repetition: Elements can be chosen multiple times (like drawing cards with replacement)