What is the difference between permutation and combination?

Permutations count ordered arrangements (ABC ≠ BCA). Combinations count unordered selections (ABC = BCA). P(n,r) = C(n,r) × r!.

When do I use "with repetition"?

When items can be selected more than once. Choosing 3 ice cream scoops from 5 flavors with repeat = C(7,3) = 35.

What is Pascal's triangle?

A triangular array where each entry C(n,r) equals the sum of the two entries above it. Row n contains all C(n,0) through C(n,n).

How do I calculate large factorials?

Use Stirling's approximation: n! ≈ √(2πn)(n/e)ⁿ. Or compute C(n,r) iteratively to avoid factorial overflow.

What are the lottery odds?

For a 6/49 lottery: C(49,6) = 13,983,816. Your chance of winning is about 1 in 14 million.

Where do combinations appear in statistics?

Binomial distribution uses C(n,k). Hypothesis testing, confidence intervals, and sampling all rely on combinatorial formulas.

Permutation & Combination Calculator

Calculate P(n,r) and C(n,r), with and without repetition, compare all four variants, visualize Pascal's triangle, and explore the computation table.

Display

n (total objects)

r (selections)

Repetition

P(n,r)

5,040

10! / (10-4)! — ordered, no repetition

C(n,r)

210

10! / (4! × (10-4)!) — unordered, no repetition

P(n,r) with rep

10,000

n^r = 10^4 — ordered, with repetition

C(n+r-1,r)

715

(10+4-1)! / (4!×(10-1)!) — unordered, with repetition

P(n,r) / C(n,r)

r! = 4! — number of orderings per combination

C(n,r)/C(n,r) Symmetry

C(10,4) = C(10,6)

Both = 210

Pascal's Triangle (rows 0–10)

121

1331

14641

15101051

1615201561

172135352171

18285670562881

193684126126843691

1104512021025221012045101

Highlighted: C(10,4) = 210

Comparison Table (n = 10)

r	P(n,r)	C(n,r)	P rep	C rep
0	1	1	1	1
1	10	10	10	10
2	90	45	100	55
3	720	120	1,000	220
4	5,040	210	10,000	715
5	30,240	252	100,000	2,002
6	151,200	210	1,000,000	5,005
7	604,800	120	10,000,000	11,440
8	1,814,400	45	100,000,000	24,310
9	3,628,800	10	1,000,000,000	48,620
10	3,628,800	1	10,000,000,000	92,378

Planning notes, formulas, and examples

About the Permutation & Combination Calculator

Permutations and combinations are the two pillars of combinatorics — the mathematics of counting, arranging, and selecting. A permutation counts the number of ways to arrange r items from n where order matters; a combination counts selections where order doesn't matter.

This calculator computes all four variants: permutations and combinations, each with and without repetition. P(n,r) = n!/(n−r)! counts ordered arrangements. C(n,r) = n!/(r!(n−r)!) counts unordered selections (binomial coefficients). With repetition, P = nʳ and C = C(n+r−1, r). The interactive Pascal's triangle highlights your chosen C(n,r), and a comparison table shows all values for every r from 0 to n.

These formulas are essential in probability (sample spaces), statistics (hypothesis testing), cryptography (key space analysis), genetics (allele combinations), algorithm analysis (complexity bounds), and everyday problems like lottery odds, committee selection, and tournament brackets. The binomial coefficient C(n,r) is also the coefficient of xʳ in (1+x)ⁿ, connecting combinatorics to algebra.

When This Page Helps

Factorials grow extremely fast — 20! already exceeds 2.4 × 10¹⁸ — making manual calculation impractical for even moderate values of n. Comparing all four variants (P and C, with and without repetition) requires multiple factorial computations and careful formula selection. This calculator computes P(n,r), C(n,r), and their repetition variants, highlights your value on Pascal's triangle, and provides a comparison table across all r values so you can see patterns at a glance. Whether you're calculating lottery odds, committee selections, or sample spaces for probability, the page cuts down arithmetic overhead and checking time.

How to Use the Inputs

Enter n (total number of objects) and r (number of selections).
Choose whether to display permutations, combinations, or both.
Select repetition mode: without, with, or compare both.
Use presets for common scenarios like poker hands (52 choose 5).
Review all four computed values in the output cards.
Check Pascal's triangle to find your C(n,r) visually.
Use the comparison table to see how values change with different r.

Formula used

P(n,r) = n! / (n−r)!
C(n,r) = n! / (r! · (n−r)!)
P rep = nʳ
C rep = C(n+r−1, r)

Example Calculation

Result: P(10,4) = 5,040; C(10,4) = 210

10 × 9 × 8 × 7 = 5,040 ordered arrangements. Dividing by 4! = 24 gives 210 unordered combinations.

Tips & Best Practices

C(n,r) = C(n, n−r) — choosing r items is equivalent to choosing which n−r to leave out.
P(n,r) = C(n,r) × r! — each combination has r! orderings.
C(n,0) = C(n,n) = 1 — there's exactly one way to choose nothing or everything.
The sum of row n in Pascal's triangle equals 2ⁿ.
For large n, use the logarithm of factorials to avoid overflow.

Permutations vs. Combinations: When Order Matters

The central question in any counting problem is: does order matter? Arranging 3 books on a shelf is a permutation problem (ABC ≠ BAC), while choosing 3 books to read is a combination problem (the set {A,B,C} is the same regardless of order). P(n,r) = C(n,r) × r!, meaning each unordered selection of r items can be arranged in r! ways. For 52-card poker: C(52,5) = 2,598,960 possible hands, but P(52,5) = 311,875,200 possible ordered deals.

Pascal's Triangle and the Binomial Theorem

Pascal's triangle is an infinite triangular array where each entry equals C(n,r). Row n contains the binomial coefficients for expanding (a+b)ⁿ. Key properties: the sum of row n equals 2ⁿ (total subsets of an n-element set), and C(n,r) = C(n−1,r−1) + C(n−1,r) gives the recursive construction. Pascal's triangle also encodes Fibonacci numbers (diagonal sums), hockey stick identities, and Catalan number relationships. It appears throughout probability, algebra, and number theory.

Real-World Applications of Combinatorics

Lottery odds use C(n,r) directly: a 6/49 lottery has C(49,6) = 13,983,816 outcomes. In genetics, C(n,r) counts genotype combinations. In networking, C(n,2) gives the number of handshakes or connections between n nodes. Committee selection from a group, pizza topping combinations, and tournament bracket arrangements are all combination problems. Cryptographic key spaces, error-correcting codes (Hamming codes use binomial coefficients), and statistical sampling distributions all rely on permutation and combination formulas.

Sources & Methodology

Last updated: January 15, 2025

Frequently Asked Questions

Permutations count ordered arrangements (ABC ≠ BCA). Combinations count unordered selections (ABC = BCA). P(n,r) = C(n,r) × r!.