Combinations Calculator C(n,r)
Calculate C(n,r) = n!/[r!(n-r)!] — ways to choose r items from n when order does not matter. Used in probability, statistics, and the binomial theorem.
🔢 Combinatorics📐 C(n,r) = n! / (r! × (n-r)!)🔢 Math
Total items n
Items chosen r
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Formula & Reference
| Variable | Symbol | Formula | Units |
|---|---|---|---|
| Combinations Calculator C(n,r) | — | C(n,r) = n! / (r! × (n-r)!) | positive integer |
Step-by-Step Examples
Example 1
Lottery
C(49,6)
- 13,983,816 combinations
- ~1 in 14 million chance
✓ 13,983,816
Example 2
Committee
C(10,3)
- 10×9×8/6=120
✓ 120
Example 3
Card hand
C(52,5)
- 2,598,960 hands
✓ 2,598,960
Real-World Applications
Probability
P = favorable/total combinations.
Stats
Core of binomial distribution.
CS
Subset counting, Pascal's triangle.
Sports
Tournament selection probabilities.
Common Mistakes to Avoid
⚠️
C vs P
C: order doesn't matter. P: order matters. C(n,r)=P(n,r)/r!
⚠️
Optimize computation
C(n,r)=C(n,n-r). Use the smaller side.
⚠️
Not dividing by r!
Forgetting r! gives permutations, not combinations.
Frequently Asked Questions
C vs P? ▾
C: choosing (same set). P: arranging (order matters).
Pascal's triangle? ▾
Row n gives C(n,0)...C(n,n). Each entry = sum of two above.
Binomial theorem? ▾
(a+b)^n=ΣC(n,k)a^(n-k)b^k.
C(n,0) and C(n,n)? ▾
Both = 1.
Complement rule? ▾
C(n,r)=C(n,n-r).
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Formula Explorer connections
Interpretation: This relationship counts discrete structures, analyzes logic and sets, or describes algorithmic growth. Assumption: Define the sample space, recurrence, logical variables and counting constraints precisely. Avoid double counting and distinguish worst, average and best cases.