Permutation of Multiset Calculator

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A -1-	➔	A -2-	➔	B -3-

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A B C	AA AB AC	AAB AAC

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Permutation of Multiset

Permutation is a mathematical concept that refers to the arrangement of elements from a collection, where the order in which the elements are chosen affects the result. In the case of a permutation of multisets, the elements are not all distinct, leading to some elements appearing multiple times.When calculating permutations for multisets, one must consider the frequency of each distinct element to ensure that identical arrangements are not counted multiple times.

Permutation of Multiset Formula

In cases where some elements in a set may be repeated, the number of possible outcomes is calculated using the permutation of multiset formula:

P = \frac{n!}{r_{1}! \times r_{2}! \times \dots \times r_{n}!}

P = Permutation | n = total number of elements | r1! x r2! x …. x rn! = the frequencies of the repeated elements

Permutation of Multiset Examples

Explore the following Permutation of Multiset examples to understand how to calculate arrangements in various scenarios.
Example 1: Permutations of a Multiset of Letters

Problem: How many different ways can you arrange the letters A, A, B, and B?
Solution: There are 4 letters, with A repeating twice and B repeating twice, 4! / 2! x 2! = 24 / 4 = 6.
Permutations: {AABB}, {ABAB}, {ABBA}, {BAAB}, {BABA}, {BBAA}.

Example 2: Permutations of a Multiset of Numbers

Problem: How many different ways can you arrange the numbers 1, 1, 2, and 3?
Solution: There are 4 numbers, with 1 repeating twice, 4! / 2! = 24 / 2 = 12.
Permutations: {1123}, {1132}, {1213}, {1231}, {1312}, {1321}, {2113}, {2131}, {2311}, {3112}, {3121}, {3211}.

Example 3: Permutations of a Multiset of Colors

Problem: How many different ways can you arrange the colors red, red, blue, and green?
Solution: There are 4 colours, with red repeating twice, 4! / 2! = 24 / 2 = 12.
Permutations: {red, red, blue, green}, {red, red, green, blue}, {red, blue, red, green}, {red, blue, green, red}, {red, green, red, blue}, {red, green, blue, red}, {blue, red, red, green}, {blue, red, green, red}, {blue, green, red, red}, {green, red, red, blue}, {green, red, blue, red}, {green, blue, red, red}.

Permutation of Multiset Exercise

Engage in this Permutation of Multiset exercise to explore the concept of permutations through practical questions. Test your ability to calculate arrangements.
Que 1: How many distinct permutations can you create with the letters {M, M, N, O}?
Ans 1: 12.
Que 2: How many different ways can you arrange the numbers {2, 2, 4, 5}?
Ans 2: 12.
Que 3: How many unique ways can you order the items {apple, apple, orange, banana}?
Ans 3: 12.
Que 4: How many ways can you arrange the set of letters {A, A, B, B, C}?
Ans 4: 30.
Que 5: How many ways can you arrange the set of numbers {1, 2, 2, 3, 3}?
Ans 5: 30.

Permutation of Multiset Calculator FAQ

In what situations should I use permutations of a multiset?

Permutations of a multiset are used when arranging elements where some elements are repeated, such as in cryptography when generating passwords with repeated characters or in scheduling when assigning tasks with identical requirements.

How does the presence of identical elements affect the total number of permutations?

The presence of identical elements decreases the total number of unique permutations compared to a set with all distinct elements.

What if all objects in the multiset are identical?

If all objects are identical, the number of permutations is simply 1, as there is only one way to arrange them.

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