Permutation with Repetition Calculator

Type:

Total Elements (n)

Selected Elements (r)

Total Elements (Individuals) [4] 𝒊

Selected Elements (Seats Available) [3] 𝒊

A -1-	➔	A -2-	➔	A -3-

Permutation Selector 𝒊


A B C D	AA AB AC AD	AAA AAB AAC AAD [#1] [#2] [#3] [#4]

Permutation with Repetition

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 permutation with repetition, elements can be selected multiple times, allowing for infinite repetition of any element in the collection. This means the same element can appear in several positions within the arrangement.

Permutation with Repetition Formula

In cases where some elements in a collection can be repeated, the number of arrangements is calculated using the permutation with repetition formula:

P = n^{r}

P = Permutation | n = total number of elements | r = number of elements to choose

Permutation with Repetition Examples

Explore the following Permutation with Repetition examples to understand how to calculate arrangements in various scenarios.
Example 1: PIN Code

Problem: How many 4-digit PIN codes can be created using the digits 0 to 9?
Solution:
- There are 10 choices (digits 0 to 9) for each of the 4 positions.
- Since each digit can be repeated, the total number of PIN codes is: 𝑛^𝑟 = 10^4 = 10000.
Answer: There are 10000 different 4-digit PIN codes.

Example 2: Tossing a Coin

Problem: A coin is tossed 3 times. How many possible outcomes are there?
Solution:
- For each coin toss, there are 2 possible outcomes: heads or tails.
- Since the coin is tossed 3 times: 𝑛^𝑟 = 2^3 =8.
Answer: There are 8 possible outcomes for the 3 coin tosses.

Example 3: Lock Combination

Problem: How many different 3-digit lock combinations are possible if each digit can be any number from 1 to 5?
Solution:
- There are 5 choices (digits 1 to 5) for each of the 3 positions.
- Since each digit can be repeated, the total number of lock combinations is: 𝑛^𝑟 = 5^3 = 125.
Answer: There are 125 different 3-digit lock combinations possible.

Permutation with Repetition Exercise

Engage in this Permutation with Repetition exercise to explore the concept of permutations through practical questions. Test your ability to calculate arrangements.
Que 1: How many 3-letter words can be formed using the letters A, B, and C with repetition?
Ans 1: 27.
Que 2: How many 2-digit numbers can be formed using the digits 1, 2, 3 with repetition?
Ans 2: 9.
Que 3: How many 4-digit PINs can be formed using the digits 0-9 with repetition?
Ans 3: 10000.
Que 4: How many ways can you arrange 4 letters from {X, Y, Z} with repetition?
Ans 4: 81.
Que 5: How many ways can you arrange 5 letters from {A, B} with repetition?
Ans 5: 32.

Permutation with Repetition Calculator FAQ

What is the difference between permutations with and without repetition?

In permutations without repetition, each element can only be used once in an arrangement. In contrast, permutations with repetition allow elements to be reused, leading to a larger number of possible arrangements.

Are there limitations to permutations with repetition?

The main limitation is that while elements can be repeated, the total number of arrangements may become impractical for large n or r, leading to extremely large numbers of permutations that can be difficult to compute or manage.

What happens in permutations with repetition when the total number of elements is less than the number of chosen elements (i.e., n < r)?

When n < r, permutations with repetition still apply because there can be multiple repetitions of items as needed according to the situation. This flexibility allows for a greater variety of arrangements, as each of the r positions can be filled with any of the n items.

Total Elements (Individuals)

This visualisation represents the collection of elements that can be repeated.
Even though elements can be repeated, each distinct element is treated as a unique choice.

Selected Elements (Seats Available)

It consists of a specific number of seats to illustrate how many elements can be selected and arranged.
Each seat is uniquely numbered to indicate the positions that elements can occupy.
The same element can occupy multiple positions since it can be repeated infinitely.
The arrows emphasise that the order in which elements are seated is crucial.
The main purpose of this visualisation is to show that the arrangement of elements matters.

Permutation Selector

This visualisation features a tree structure.
Each level of the tree contains all the arrangements for the selected elements, where the number of selected elements is equal to the level number.
The final level displays the desired outcome along with its entry number.
Dropdown menus let you select elements interactively, making it simple for you to explore different arrangements.
Previous levels act as navigational tools leading to this outcome.
This design enables easy step-by-step exploration of arrangements, allowing for backtracking.