Permutation & Combination Calculator

Calculate square roots, cube roots, and more.

The total number of distinct items in the set
The number of items to be selected from the set
Result

0

Formula Used

How to Use the Calculator

1
Select Calculation Type

Choose between Permutation (nPr) or Combination (nCr) based on your needs:

  • Permutation (nPr): Use when the order of selection matters
  • Combination (nCr): Use when the order of selection doesn't matter
2
Enter Total Items (n)

Enter the total number of distinct items in your set. This must be a non-negative integer.

3
Enter Selected Items (r)

Enter the number of items you want to select from the set. This must be a non-negative integer and cannot exceed the total number of items.

4
Click Calculate

Click the "Calculate" button to get your result. The calculator will display the result along with the formula used.

Examples

Example 1: Permutation

Problem: In how many ways can 3 people be selected from 8 people and arranged in a line?

Solution: This is a permutation problem because the order matters (different arrangements in line).

  • Total items (n) = 8
  • Selected items (r) = 3
  • Formula: 8P3 = 8!/(8-3)! = 8!/5! = 336
Example 2: Combination

Problem: In how many ways can a committee of 4 people be formed from 10 people?

Solution: This is a combination problem because the order doesn't matter (just who is on the committee).

  • Total items (n) = 10
  • Selected items (r) = 4
  • Formula: 10C4 = 10!/(4!(10-4)!) = 10!/(4!6!) = 210

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Permutations and Combinations

What are Permutations and Combinations?

Permutations and combinations are fundamental concepts in combinatorics, a branch of mathematics dealing with counting and arrangement. They help us determine the number of ways to select or arrange objects under specific conditions.

Permutations refer to the number of ways to arrange a set of objects where the order matters. For example, arranging the letters A, B, and C in different orders (ABC, BAC, CBA) counts as different permutations.

Combinations refer to the number of ways to select a subset of objects where the order does not matter. For example, selecting 2 letters from A, B, and C (e.g., AB and BA are the same combination).

Formulas for Permutations and Combinations

Permutations

The number of permutations of \( n \) distinct objects taken \( r \) at a time is given by:

\( P(n, r) = \frac{n!}{(n - r)!} \)

Where \( n! \) (n factorial) is the product of all positive integers up to \( n \) (e.g., \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)).

Combinations

The number of combinations of \( n \) distinct objects taken \( r \) at a time is given by:

\( C(n, r) = \frac{n!}{r! \times (n - r)!} \)

The extra \( r! \) in the denominator accounts for the fact that order does not matter in combinations.

Process to Calculate Permutations and Combinations

Calculating Permutations

1
Identify \( n \) and \( r \): Determine the total number of objects (\( n \)) and how many you are arranging (\( r \)).
2
Compute Factorials: Calculate \( n! \) and \( (n - r)! \).
3
Apply the Formula: Use \( P(n, r) = \frac{n!}{(n - r)!} \).

Calculating Combinations

1
Identify \( n \) and \( r \): Determine the total number of objects (\( n \)) and how many you are selecting (\( r \)).
2
Compute Factorials: Calculate \( n! \), \( r! \), and \( (n - r)! \).
3
Apply the Formula: Use \( C(n, r) = \frac{n!}{r! \times (n - r)!} \).

Examples: Calculating Permutations and Combinations

Permutations Example

Example: How many ways can you arrange 3 letters from the set {A, B, C, D}?
- Step 1: Here, \( n = 4 \) (total letters), \( r = 3 \) (letters to arrange).
- Step 2: Compute factorials: \( 4! = 4 \times 3 \times 2 \times 1 = 24 \), \( (4 - 3)! = 1! = 1 \).
- Step 3: Apply the formula: \( P(4, 3) = \frac{4!}{(4 - 3)!} = \frac{24}{1} = 24 \).
So, there are 24 ways to arrange 3 letters from {A, B, C, D}.

Combinations Example

Example: How many ways can you select 2 letters from the set {A, B, C, D}?
- Step 1: Here, \( n = 4 \) (total letters), \( r = 2 \) (letters to select).
- Step 2: Compute factorials: \( 4! = 24 \), \( 2! = 2 \times 1 = 2 \), \( (4 - 2)! = 2! = 2 \).
- Step 3: Apply the formula: \( C(4, 2) = \frac{4!}{2! \times (4 - 2)!} = \frac{24}{2 \times 2} = \frac{24}{4} = 6 \).
So, there are 6 ways to select 2 letters from {A, B, C, D} (e.g., AB, AC, AD, BC, BD, CD).

Why Should Permutations and Combinations Be Used?

Permutations and combinations are essential tools in mathematics and related fields because they help solve problems involving counting and arrangement efficiently. Here are some key reasons to use them:

  • Probability and Statistics: They are used to calculate probabilities in scenarios like lottery draws or card games, where the number of possible outcomes matters.
  • Computer Science: Algorithms for sorting, searching, or generating arrangements (e.g., passwords, schedules) often rely on permutation and combination principles.
  • Decision Making: In business or logistics, they help determine the number of possible team formations, project assignments, or resource allocations.
  • Education and Research: They provide a foundation for understanding more complex mathematical concepts, such as binomial theorem or graph theory.
  • Real-Life Applications: From designing seating arrangements to creating sports tournament brackets, permutations and combinations offer practical solutions to arrangement and selection problems.

By understanding and applying permutations and combinations, you can tackle a wide range of problems with precision, making them invaluable tools in both theoretical and practical contexts.

Frequently Asked Questions (FAQ)

The key difference is that permutations consider the order of arrangement (e.g., ABC and BAC are different), while combinations do not (e.g., AB and BA are the same). This is why the combination formula includes an extra \( r! \) in the denominator to eliminate duplicate arrangements.

If \( r > n \), the permutation or combination is not possible because you cannot select or arrange more items than are available. Mathematically, the formula will result in a denominator larger than the numerator, yielding zero or an undefined result.

Factorials represent the number of ways to arrange objects. For example, \( n! \) gives the total arrangements of \( n \) distinct objects. In permutations and combinations, factorials help calculate the total possibilities and adjust for constraints like order or selection size.

In probability, they help calculate the total number of possible outcomes. For example, to find the probability of drawing 2 aces from a deck of 52 cards, you use combinations: \( C(4, 2) \) for the favorable outcomes (4 aces, choose 2) and \( C(52, 2) \) for the total outcomes (52 cards, choose 2).