Combination without Repetition Calculator
Calculate combinations without repetition using the formula C(n,r) = n! / (r!(n-r)!)
Calculate Combinations without Repetition
Total distinct objects available for selection
Number of objects to choose (r ≤ n)
Combination Results
Formula: C(n,r) = n! / (r!(n-r)!)
Where: n = 0, r = 0
Example Calculation
4 Numbers from 0-9 Example
Problem: How many combinations of 4 numbers from decimal system (0-9)?
Total objects (n): 10 (digits 0-9)
Sample size (r): 4
Solution
C(10,4) = 10! / (4!(10-4)!)
C(10,4) = 10! / (4! × 6!)
C(10,4) = (10×9×8×7) / (4×3×2×1)
C(10,4) = 5040 / 24
C(10,4) = 210
Key Concepts
Combinations vs Permutations
- Combinations: Order doesn't matter
- Permutations: Order matters
- Example: {1,2,3} = {3,2,1} in combinations
Without Repetition
- Each object can be selected only once
- No duplicates in combinations
- r cannot exceed n
Formula Properties
- C(n,0) = 1
- C(n,n) = 1
- C(n,1) = n
C(n,r) = C(n-1,r-1) + C(n-1,r)
Quick Tips
Use when order doesn't matter
Each object selected only once
Sample size must be ≤ total objects
Result is always a positive integer
Understanding Combinations without Repetition
What are Combinations without Repetition?
Combinations without repetition are selections of objects from a larger set where:
- •The order of selection doesn't matter
- •Each object can only be selected once
- •All objects in the set are distinct
Real-World Applications
- •Selecting team members from a group
- •Choosing lottery numbers
- •Menu combinations at restaurants
- •Selecting samples for research
The Combination Formula
C(n,r) = n! / (r!(n-r)!)
- C(n,r): Number of combinations
- n: Total number of objects
- r: Number of objects to choose
- n!: n factorial (n × (n-1) × ... × 1)
Why This Formula Works
The formula accounts for all possible arrangements (n!), then divides by the arrangements within the chosen group (r!) and the arrangements within the remaining group ((n-r)!) since order doesn't matter in combinations.
Note: For large values of n, this calculator uses an optimized method to avoid computing large factorials directly.