Permutation & Combination Calculator
Calculate permutations nPr and combinations nCr instantly. Includes factorial table, Pascal's triangle, circular permutations, multiset permutations, and combinations with repetition.
Quick Calculator Get a fast estimate
nCr — Combinations C(10,3)
120
nPr — Permutations P(10,3)
720
n!
3 628 800
r!
6
(n-r)!
5 040
nPr / nCr ratio
6 (= r!)
Link copied to clipboard!
How to Use the Permutation & Combination Calculator
Enter n (total items) and r (how many to choose/arrange). Both nCr and nPr are computed simultaneously.
Use combinations when order does not matter (choosing a committee). Use permutations when order matters (arranging a race finish). The ratio nPr/nCr = r!.
Need more detail?
Extended Calculator More options, charts, and scenario comparison
Combinations C(10,3)
120
Formula
10! / (3!×7!)
nCr
120
nPr
720
nPr = nCr × r!
120 × 6 = 720
Pascal's Triangle (C(n,0)...C(n,n) for row n)
1
11
121
1331
14641
15101051
1615201561
172135352171
18285670562881
Permutation & Combination Formulas
nPr = n! / (n−r)! (ordered arrangements)
nCr = n! / (r! × (n−r)!) (unordered selections)
nPr = nCr × r!
Circular permutations: (n−1)!
Permutations with repetition: nʳ
Combinations with repetition: C(n+r−1, r)
Multinomial: n! / (k₁!×k₂!×...×kₘ!)
Common Examples
| Problem | Formula | Answer |
|---|---|---|
| Choose 3 from 10 (order irrelevant) | C(10,3) | 120 |
| Arrange 3 from 10 (order matters) | P(10,3) | 720 |
| 8 people around a table | (8−1)! | 5,040 |
| 4-digit PIN (digits 0-9, repeat ok) | 10⁴ | 10,000 |
| Lottery: 6 from 49 | C(49,6) | 13,983,816 |
Need full precision?
Professional Calculator Complete parameters, sensitivity analysis, and detailed breakdown
Multinomial coefficient: n! / (k₁!×k₂!×...)
Group sizes (must sum to n=6)
k1:
k2:
k3:
Multinomial Coefficient
60
6! / (3! × 2! × 1!) = 720 / 12
Frequently Asked Questions
Permutation: order matters (selecting president, VP, treasurer from 10 people = P(10,3)=720). Combination: order doesn't matter (selecting a 3-person committee from 10 = C(10,3)=120). Always P ≥ C.
nCr = n! / (r! × (n−r)!). Also written C(n,r) or "n choose r". It appears in Pascal's triangle and the binomial theorem: (a+b)ⁿ = Σ C(n,k)·aⁿ⁻ᵏ·bᵏ.
nPr = n! / (n−r)!. You can also compute it as nCr × r! — first choose the items (nCr ways), then arrange them (r! ways). Example: P(6,2) = 6×5 = 30.
(n−1)! arrangements. Seating 5 people at a round table: (5−1)! = 24 ways (not 5!=120) because rotating everyone one seat gives the same arrangement.
Arranging n objects with repetitions: n!/(k₁!k₂!...kₘ!). Example: how many ways to arrange MISSISSIPPI (11 letters: 1M, 4I, 4S, 2P)? 11!/(1!4!4!2!) = 34,650.