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Permutation and Combination Calculator
Easily calculate permutations (nPr) and combinations (nCr) with our free online calculator. Perfect for students, teachers, and professionals working with probability, statistics, and combinatorics.
Understanding Permutations and Combinations
Permutations and combinations are fundamental concepts in mathematics that deal with arranging and selecting objects from a set.
What are Permutations (nPr)?
Permutations refer to the number of ways to arrange 'r' objects from a set of 'n' distinct objects where order matters. The formula for permutations is:
nPr = n! / (n - r)!
What are Combinations (nCr)?
Combinations refer to the number of ways to select 'r' objects from a set of 'n' distinct objects where order does not matter. The formula for combinations is:
nCr = n! / [r! × (n - r)!]
Practical Applications
These concepts are widely used in probability theory, statistics, computer science, and various real-world scenarios like:
- Lottery probability calculations
- Password combination possibilities
- Committee formation possibilities
- Game theory scenarios
- Genetic combination probabilities
Frequently Asked Questions
What is the difference between permutation and combination?
The key difference is that order matters in permutations but not in combinations. For example, the arrangement ABC is different from CBA in permutations, but they are considered the same in combinations.
Can n be smaller than r in these calculations?
No, for both nPr and nCr, the value of n must be greater than or equal to r. If r is greater than n, the result is undefined (or zero by convention in combinatorics).
What is factorial notation?
Factorial notation (n!) represents the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition, 0! = 1.
What are some real-world examples of permutations?
Real-world examples include: arranging books on a shelf, creating passwords with different character orders, determining race finishing orders, and scheduling tasks in sequence.
What are some real-world examples of combinations?
Common examples include: selecting team members from a group, choosing lottery numbers, picking menu items from a list, forming committees, and selecting toppings for pizza.
