Combinations and Permutations
Calculate combinations (nCr) and permutations (nPr) for any values.
What Is the Combinations and Permutations?
A combinations and permutations calculator answers counting questions where order matters or does not. Permutations count arrangements: three podium finishes from ten runners is 10 pick 3 because placement order matters. Combinations count selections: choosing a three person committee from ten people is 10 choose 3 since membership alone counts. The two differ by a factor of the factorial of the selection size, which is why permutation counts always run higher. Lottery odds, poker hands, password strength, seating charts, and quality sampling all reduce to these two formulas, and this tool computes them exactly with full working.
Key Statistics
-
13,983,816
Combinations in a six number lottery draw from 49 balls
Source: Standard lottery mathematics
-
720
Ordered arrangements of ten items taken three at a time
Source: Permutation formula result
-
n choose k
Binomial coefficient notation connecting counting to algebra through Pascal's triangle
Source: Combinatorics notation
The Formula
nCr = n! / (r! x (n-r)!). nPr = n! / (n-r)!.
Worked Examples
Choosing 3 committee members from 10 candidates gives 120 combinations where order does not matter. Arranging 3 books from 10 on a shelf yields 720 permutations since each position is distinct. These formulas power poker hand analysis, tournament bracket design, and lottery probability calculations.
Committee selection
- Choose 2 members from 5 candidates where order is irrelevant
- Apply 5 factorial over 2 factorial times 3 factorial
- Compute 120 over 4
There are 10 possible committees.
Podium outcomes
- Rank gold, silver, bronze among 10 racers
- Order matters, so apply permutations: 10 pick 3
- Compute 10 times 9 times 8
There are 720 distinct podium orders.
Real World Use Cases
Lottery and gambling odds
Compute exact jackpot odds before spending on tickets.
Event planning
Count seating arrangements, gift bag combinations, or schedule permutations.
Quality control
Determine how many sample subsets exist for inspection planning.
Expert Tips
- ✓ Ask one question first: would swapping two chosen items produce a different outcome? Yes means permutations.
- ✓ Combinations never exceed permutations; the ratio between them is always the factorial of k.
- ✓ For repeated items, multinomial counting extends these formulas rather than replacing them.
- ✓ Pascal's triangle gives small binomial coefficients by simple addition, handy without a calculator.
Frequently Asked Questions
What is the difference between combinations and permutations? ▼
Order matters for permutations and does not matter for combinations. Selecting A then B versus B then A counts as two permutations but only one combination. A lock code is a permutation; a lottery draw is a combination.
How do I calculate combinations? ▼
Use the formula nCr equals n! divided by (r! times (n minus r)!). Choosing 3 players from a team of 10 is 10! divided by (3! times 7!), which is 120 ways.
How do I calculate permutations? ▼
Use the formula nPr equals n! divided by (n minus r)!. Arranging 3 books from 5 in order is 5! divided by 2!, which is 60 ways.
When should I use combinations versus permutations? ▼
Ask whether the order of the result matters. Seating arrangements and rankings are permutations. Committees, card hands, and lotto numbers are combinations, because reordering does not change the selection.
What does nCr mean? ▼
It reads as n choose r and counts how many ways you can pick r items from a set of n, ignoring order. The value 5 choose 2 equals 10, meaning 10 ways to pick 2 items from 5.
How many ways can I arrange n items? ▼
Arranging all n items in order gives n! possibilities. Three books can be arranged 3!, or 6 ways, and four people can line up 4!, or 24 ways. This calculator counts the exact total for you.
What is the difference between combination and permutation? ▼
Permutations count ordered arrangements while combinations count unordered selections, making permutation counts larger whenever order matters.
How do I calculate combinations? ▼
Divide n factorial by the product of k factorial and the quantity n minus k factorial.
Common Mistakes to Avoid
- ⚠ Choosing combinations when order matters or permutations when it does not. Lottery numbers are combinations (order irrelevant), but podium finishes are permutations (gold, silver, bronze positions are distinct). Using the wrong mode gives an incorrect count.
- ⚠ Entering r larger than n. You cannot select 10 items from a set of 5. The calculator requires r to be less than or equal to n and will not compute invalid selections.
- ⚠ Forgetting that nCr is symmetric (nCr = nC(n-r)) while nPr is not. C(10, 8) equals C(10, 2) = 45, but P(10, 8) is vastly larger than P(10, 2). Expecting symmetry for permutations leads to wildly wrong counts.
- ⚠ Using combinations where order matters. A race result counted with combinations undercounts outcomes massively.
Related Tools & Comparisons
Last updated: · by CalculatorPro Tools