Permutations & Combinations
Calculate nPr (permutations) and nCr (combinations) with large number support
Total number of items (max 170)
Number of items to select (r ≤ n)
Permutations (nPr)
Permutations count the number of ways to arrange r items from a set of n items where order matters.
P(n, r) = n! / (n-r)!
Example: How many ways can 3 people finish a race from 10 runners? P(10, 3) = 720
Combinations (nCr)
Combinations count the number of ways to choose r items from a set of n items where order does not matter.
C(n, r) = n! / (r! × (n-r)!)
Example: How many ways can you choose 2 cards from 5? C(5, 2) = 10
Key Difference
- Permutation: Order matters (ABC ≠ BAC)
- Combination: Order doesn't matter (ABC = BAC)
- P(n, r) is always ≥ C(n, r) because permutations count more arrangements
Applications: Probability theory, statistics, lottery calculations, team selection, password combinations, and cryptography.
Frequently Asked Questions
What's the difference between permutations and combinations?
Permutations count arrangements where order matters (ABC is different from BAC), while combinations count selections where order doesn't matter (ABC equals BAC). Use permutations for rankings, sequences, or arrangements. Use combinations for groups, teams, or selections where position doesn't matter.
When should I use permutations vs combinations?
Use permutations (nPr) when: assigning positions in a race, creating passwords, arranging books on a shelf. Use combinations (nCr) when: choosing lottery numbers, selecting team members, picking cards from a deck. Ask yourself: 'Does the order of selection change the outcome?' If yes, use permutations.
How do I calculate combinations without a calculator?
Use the formula C(n,r) = n! / (r!(n-r)!), but simplify first. For C(10,3): write 10x9x8 / 3x2x1, then cancel common factors. For C(n,r), you only need to multiply r consecutive numbers starting from n, then divide by r!. Also remember C(n,r) = C(n,n-r), so C(100,98) = C(100,2) = 4950.
What is Pascal's Triangle and how does it relate to combinations?
Pascal's Triangle displays all combination values: the entry in row n, position r equals C(n,r). Each number is the sum of the two numbers above it. Row 4 contains 1,4,6,4,1 representing C(4,0) through C(4,4). The triangle reveals patterns like the binomial theorem and has applications in probability, algebra, and combinatorics.