MathApril 12, 2026

Combination Calculator Guide: Combinations vs Permutations (2026)

Q: What is the difference between a combination and a permutation?

Combinations are selections where order doesn't matter — choosing players for a team (AB = BA). Permutations are arrangements where order matters — assigning first and second place (AB ≠ BA). For the same n and r, permutations always outnumber combinations by a factor of r! (r factorial).

Q: What is the combination formula (nCr)?

C(n, r) = n! / (r! × (n − r)!), where n is the total items and r is the number chosen. For example, choosing 3 people from 10: C(10, 3) = 10! / (3! × 7!) = 720 / 6 = 120 possible groups.

Q: What are the odds of winning the lottery?

For Powerball (choose 5 from 69, then 1 from 26): C(69,5) × 26 = 292,201,338 possible outcomes, so odds are about 1 in 292 million. For Mega Millions (5 from 70, 1 from 25): about 1 in 302 million. You're roughly 20,000 times more likely to be struck by lightning in your lifetime.

Q: What is Pascal's triangle?

Pascal's triangle is a triangular array where each number is the sum of the two numbers above it. Row n contains the values C(n,0), C(n,1), ..., C(n,n). For example, row 4 is 1, 4, 6, 4, 1 — giving the values of C(4,0) through C(4,4). It's useful for binomial coefficients, probability, and combinatorial identities.

Q: What is a combination with repetition?

Combinations with repetition allow choosing the same item more than once. The formula is C(n + r − 1, r) = (n + r − 1)! / (r! × (n − 1)!). Example: choosing 3 scoops from 5 ice cream flavors (repeats allowed) = C(7, 3) = 35 possible selections.

By The hakaru Team·Last updated March 2026

Quick Answer

*Combination: order doesn't matter. C(n,r) = n! / (r! × (n−r)!)
*Permutation: order matters. P(n,r) = n! / (n−r)!
*Quick test: “Does rearranging the selection create a new outcome?” Yes → permutation. No → combination.
*Lottery odds use combinations: Powerball odds are about 1 in 292 million.

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Combinations vs. Permutations

The key question: does order matter?

Choosing 3 pizza toppings from 8: Order doesn't matter → combination.
Assigning president, VP, and treasurer from 8 candidates: Order matters → permutation.

From 8 items choosing 3: C(8,3) = 56 combinations vs. P(8,3) = 336 permutations. Permutations always exceed combinations by r! (here, 3! = 6, and 56 × 6 = 336).

The Combination Formula

C(n, r) = n! / (r! × (n − r)!)

Worked Example

A committee of 4 from 12 people: C(12, 4) = 12! / (4! × 8!) = (12 × 11 × 10 × 9) / (4 × 3 × 2 × 1) = 11,880 / 24 = 495.

Lottery Odds

Powerball: pick 5 numbers from 1–69 (combination) then 1 Powerball from 1–26.

White ball combinations: C(69, 5) = 11,238,513
Total outcomes: 11,238,513 × 26 = 292,201,338
Odds of jackpot: 1 in 292.2 million

Pascal's Triangle

Each entry in Pascal's triangle equals C(n, r). Row 5: 1, 5, 10, 10, 5, 1. These are C(5,0) through C(5,5). The triangle reveals patterns: symmetry (C(n,r) = C(n, n−r)), each row sums to 2ⁿ, and the entries are binomial coefficients used in expanding (a + b)ⁿ.

Combinations with Repetition

When you can pick the same item multiple times: C(n + r − 1, r)

Choosing 3 scoops from 5 flavors (can repeat): C(5 + 3 − 1, 3) = C(7, 3) = 35.

Applications

Probability: The number of favorable combinations divided by total combinations gives probability.
Poker hands: Total 5-card hands = C(52, 5) = 2,598,960. Full house combinations = 3,744 (probability ≈ 0.14%).
Genetics: Allele combinations in offspring follow combinatorial rules.
Software testing: Pairwise testing uses combinations to cover input parameter interactions.

Calculate combinations and permutations

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Frequently Asked Questions

What is the difference between a combination and a permutation?

Combinations: order doesn't matter (choosing team members). Permutations: order matters (assigning roles). For same n and r, permutations = combinations × r!.

What is the combination formula (nCr)?

C(n, r) = n! / (r! × (n−r)!). Choosing 3 from 10: C(10,3) = 120.

What are the odds of winning the lottery?

Powerball: about 1 in 292 million. Mega Millions: about 1 in 302 million. You're 20,000 times more likely to be struck by lightning.

What is Pascal's triangle?

A triangular array where each number is the sum of the two above. Row n gives C(n,0) through C(n,n). Useful for binomial coefficients and probability.

What is a combination with repetition?

When you can pick the same item more than once: C(n+r−1, r). Choosing 3 scoops from 5 flavors = C(7,3) = 35.