Permutation And Combination Calculator
Calculate permutations (nPr) and combinations (nCr) for a set of items. An essential tool for probability, statistics, and combinatorics.
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Permutations vs. Combinations: What's the Difference?
The Permutation and Combination Calculator is a tool from the field of combinatorics that helps you find the number of ways to choose or arrange items from a set. The key difference between the two lies in whether the order of selection matters. This calculator allows you to compute both, making it easy to solve complex probability and counting problems.
How to Use the Calculator
- Enter the Total Number of Items (n): This is the total size of the set you are choosing from.
- Enter the Number of Items to Choose (r): This is the size of the subset you are selecting. Note that 'r' cannot be greater than 'n'.
- View the Results: The calculator instantly provides both the number of possible Permutations and the number of possible Combinations.
Core Concepts Explained
Permutation (nPr): Order Matters
A permutation is an arrangement of items where the order of selection is important. For example, if you are choosing the 1st, 2nd, and 3rd place winners in a race, the order matters. A finishing order of (Alice, Bob, Charlie) is different from (Bob, Alice, Charlie).
The Formula for Permutations is:
nPr = n! / (n - r)!
Where '!' denotes the factorial (e.g., 5! = 5 × 4 × 3 × 2 × 1).
Combination (nCr): Order Does Not Matter
A combination is a selection of items where the order of selection is not important. For example, if you are choosing a committee of 3 people from a group of 10, choosing (Alice, Bob, Charlie) is the same as choosing (Bob, Alice, Charlie) because the resulting committee is identical.
The Formula for Combinations is:
nCr = n! / [r! * (n - r)!]
Practical Example
Let's say you have 5 books (n=5) and you want to know how many ways you can choose 3 of them (r=3).
- Permutation (Arrangement on a shelf): The order matters.
5P3 = 5! / (5-3)! = 120 / 2 = 60 different arrangements.
- Combination (Group to read): The order you pick them in doesn't matter.
5C3 = 5! / [3! * (5-3)!] = 120 / (6 * 2) = 10 different groups of books.
As you can see, there are always more (or equal) permutations than combinations for the same n and r.
Frequently Asked Questions (FAQ)
- When should I use permutation vs. combination?
- Ask yourself: "Does the order matter?" If yes, use permutation (e.g., passwords, race results, arranging items). If no, use combination (e.g., lottery numbers, picking a team, card hands).
- What is a factorial (!)?
- A factorial is the product of all positive integers up to that number. For example, 4! = 4 × 3 × 2 × 1 = 24. The value of 0! is defined as 1.
- Can 'r' be larger than 'n'?
- No. You cannot choose more items than are available in the total set. The calculator will not produce a result if r > n.
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