Combination Calculator
Find the number of ways to choose items where order doesn't matter. Calculate nCr (combinations) for any combination of n total items and r items to choose.
Result
- n (Total Items)
- -
- r (Items to Choose)
- -
- Combinations (nCr)
- 0
Formula & Guide
Formula
C
Combination Formula
nCr = n! / (r! × (n-r)!)
Factorial of n divided by product of r! and (n-r)!
=
Using Permutations
nCr = nPr / r!
Permutations divided by r factorial
Formula Variables
n
Total Items
The total number of items available to choose from
r
Items to Choose
The number of items to choose (must be ≤ n)
nCr
Combinations
The number of ways to choose r items from n total items where order doesn't matter
!
Factorial
The product of all positive integers up to that number
Step-by-Step Scenario
Example Scenario
n (Total Items)
5
r (Items to Choose)
3
1
Understand the Problem
- We want to choose 3 items from 5 total items, where order doesn't matter.
For example, choosing 3 people from 5 candidates for a committee (positions don't matter)
2
Apply the Formula
- 5C3 = 5! / (3! × (5-3)!)
- 5C3 = 5! / (3! × 2!)
- 5C3 = 120 / (6 × 2)
- 5C3 = 120 / 12
Additional Examples
Lottery Selection
n: 49 numbers
r: 6 to choose
Combinations
49C6 = 13,983,816
Meaning
13.9 million possible combinations
Committee Selection
n: 10 people
r: 4 members
Combinations
10C4 = 210
Meaning
210 ways to form the committee
Characteristics of Combinations
Order Doesn't Matter
Combinations count selections where order is irrelevant. ABC is the same as CBA, so they count as one combination.
Fewer Than Permutations
For the same n and r, combinations always give fewer arrangements than permutations because order doesn't matter in combinations.
Practical Applications
Used in lottery calculations, committee selection, team formation, sampling, and any scenario where only the selection matters.
Symmetry Property
nCr = nC(n-r). Choosing r items is equivalent to leaving (n-r) items, so 10C3 = 10C7 = 120.
Important Notes
- Order doesn't matter in combinations: ABC is the same as CBA, so they count as one combination.
- r cannot be greater than n. You cannot choose more items than you have available.
- Combinations are always less than or equal to permutations for the same n and r values.
- The symmetry property nCr = nC(n-r) can simplify calculations. For example, 20C18 = 20C2.
- Use combinations when only the selection matters, not the arrangement, such as choosing teams, committees, or lottery numbers.
Frequently Asked Questions
Find answers to common questions about combination calculations.