Permutation & Combination Calculator

Calculate arrangements (nPr) and selections (nCr) with step-by-step solutions

Permutation Calculator (nPr)

Calculate the number of ways to arrange r items from n total items where order matters.

Combination Calculator (nCr)

Calculate the number of ways to select r items from n total items where order doesn't matter.

Permutation & Combination with Repetition

Factorial Calculator

Calculate n! (n factorial) for any positive integer.

Circular Permutation Calculator

Calculate arrangements in a circle where rotations are considered identical.

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Understanding Permutations and Combinations: A Complete Guide

Permutations and combinations are fundamental concepts in combinatorics that help us count arrangements and selections. Our comprehensive calculator provides instant results for various types of permutation and combination problems with detailed step-by-step explanations.

What are Permutations and Combinations?

Both permutations and combinations deal with selecting items from a group, but they differ in whether the order of selection matters:

Permutations (nPr)

A permutation is an arrangement of objects where the order is important. The number of permutations of r objects selected from n objects is denoted as nPr or P(n,r).

Permutation Formula:
nPr = n! / (n-r)!

When to Use Permutations

Example: How many ways can 3 people be selected from 10 people to fill the positions of president, vice-president, and secretary?
Solution: 10P3 = 10! / (10-3)! = 10! / 7! = 10 × 9 × 8 = 720 ways

Combinations (nCr)

A combination is a selection of objects where the order doesn't matter. The number of combinations of r objects selected from n objects is denoted as nCr or C(n,r).

Combination Formula:
nCr = n! / (r! × (n-r)!)

When to Use Combinations

Example: How many ways can 3 people be selected from 10 people to form a committee?
Solution: 10C3 = 10! / (3! × 7!) = (10 × 9 × 8) / (3 × 2 × 1) = 720 / 6 = 120 ways

Factorials

A factorial (denoted as n!) is the product of all positive integers from 1 to n. Factorials are essential for calculating permutations and combinations.

Factorial Formula:
n! = n × (n-1) × (n-2) × ... × 2 × 1
Special case: 0! = 1

Common Factorial Values

Permutations and Combinations with Repetition

Permutations with Repetition

When items can be repeated in arrangements, the formula changes:

Permutation with Repetition:
Number of arrangements = n^r
where n = number of choices for each position, r = number of positions
Example: How many 3-digit codes can be made using digits 0-9 with repetition allowed?
Solution: 10^3 = 1,000 codes

Combinations with Repetition

When selecting items with repetition allowed:

Combination with Repetition:
C(n+r-1, r) = (n+r-1)! / (r! × (n-1)!)
Example: How many ways can you choose 3 scoops of ice cream from 5 flavors with repetition allowed?
Solution: C(5+3-1, 3) = C(7,3) = 35 ways

Circular Permutations

In circular arrangements, rotations are considered identical, so we fix one object's position to avoid counting rotations as different arrangements.

Circular Permutation Formula:
(n-1)! arrangements for n objects in a circle
Example: How many ways can 6 people sit around a circular table?
Solution: (6-1)! = 5! = 120 arrangements

Key Differences: Permutations vs Combinations

Aspect Permutation Combination
Order Matters Doesn't matter
Formula nPr = n!/(n-r)! nCr = n!/(r!(n-r)!)
Result Generally larger Generally smaller
Example ABC, ACB, BAC are different ABC, ACB, BAC are the same

Real-World Applications

Business and Management

Technology and Computer Science

Sports and Games

Science and Research

Advanced Concepts

Permutations with Identical Objects

When some objects are identical, we divide by the factorial of the number of identical objects:

Formula with Identical Objects:
n! / (n₁! × n₂! × ... × nₖ!)
where n₁, n₂, ..., nₖ are the counts of identical objects

Multinomial Coefficients

Extension of combinations for multiple groups:

Multinomial Formula:
n! / (n₁! × n₂! × ... × nₖ!)

Derangements

Permutations where no object appears in its original position:

Derangement Formula:
!n = n! × Σ(k=0 to n) (-1)ᵏ/k!

Problem-Solving Strategies

Step-by-Step Approach

  1. Identify the Problem Type: Does order matter?
  2. Determine Values: What are n and r?
  3. Check for Special Cases: Repetition? Restrictions?
  4. Apply the Formula: Use appropriate formula
  5. Verify the Answer: Does it make logical sense?

Common Problem Types

Tips for Success

  1. Understand the Difference: Master when to use permutations vs combinations
  2. Practice with Examples: Work through various problem types
  3. Check Your Work: Verify answers make sense in context
  4. Use Symmetry: nCr = nC(n-r) can simplify calculations
  5. Break Down Complex Problems: Divide into smaller, manageable parts

Common Mistakes to Avoid

Using Our Calculator

Our Permutation & Combination Calculator offers:

Conclusion

Understanding permutations and combinations is essential for solving counting problems in mathematics, statistics, computer science, and many real-world applications. Whether you're calculating the number of ways to arrange a sequence, select a committee, or design an experiment, our calculator provides the tools and explanations you need to solve these problems accurately and efficiently.