Combination with Repetition Calculator

Type:

Total Elements (n)

Selected Elements (r)

Total Elements (Individuals) [4] 𝒊

Selected Elements (All on same stage) [3] 𝒊

AAA

Combination Selector 𝒊


A	➔	AA	➔	AAA[#1] AAB[#2] AAC[#3] AAD[#4]

Combination with Repetition

Combination is a mathematical concept that refers to the selection of elements from a collection, where the order of the elements does not affect the result. In combination with repetition, elements can be selected more than once, allowing for repeated choices of the same element. The repetition can occur a finite or even infinite number of times, depending on the context. This type of combination is useful in scenarios where duplicates are allowed in the selection.

Combination with Repetition Formula

In cases where we want to select elements from a group where repetitions are allowed, we can determine the number of possible combinations using the combination with repetition formula:

C = \frac{(n + r - 1)!}{r! \cdot (n - 1)!}

C = Combination | n = total number of elements | r = number of elements to choose

Combination with Repetition Examples

Explore the following Combination with Repetition examples to learn how to find different ways to choose items in various contexts.
Example 1: Combinations with Repetition of Candies

Problem: How many ways can 3 candies be chosen from 5 different types, if repetition is allowed?
Solution: Using combination with repetition Formula: 7! / [3! × (7-3)!] = 7! / 3! × 4! = 35.
Answer: There are 35 ways to choose the candies.

Example 2: Combinations with Repetition of Ice Cream Flavors

Problem: How many ways can 4 scoops of ice cream be selected from 3 different flavors, if repetition is allowed?
Solution: Using combination with repetition Formula: 6! / [4! × (6 - 4)!] = 6! / 4! × 2! = 15.
Answer: There are 15 ways to select the ice cream scoops.

Example 3: Combinations with Repetition of Coins

Problem: How many ways can 6 identical coins be distributed among 4 different jars?
Solution: Using combination with repetition Formula: 9! / [6! × (9-6)!] = 9! / (6! × 3!) = 84.
Answer: There are 84 ways to distribute the coins.

Combination with Repetition Exercise

Engage in this Combination with Repetition exercise to explore the concept of combinations through practical questions. Challenge your skills in determining how to select items.
Que 1: How many ways can 3 candies be chosen from 5 different types, if repetition is allowed?
Ans 1: 35.
Que 2: How many ways can 4 fruits be selected from 6 different types, if repetition is allowed?
Ans 2: 126.
Que 3: How many ways can 5 students be chosen from 8 different classes, if a student can be chosen more than once?
Ans 3: 792.
Que 4: How many ways can 2 marbles be selected from 4 different colors, if each color can be chosen more than once?
Ans 4: 10.
Que 5: How many ways can 7 identical coins be distributed among 3 children?
Ans 5: 36.

Combination with Repetition Calculator FAQ

What is the difference between combinations and combinations with repetition?

Combinations involve selecting items where each item can only be chosen once, while combinations with repetition allow items to be selected multiple times.

What does C(n, r) mean when r > n in combinations with repetition?

C(n, r) in combination with repetition allows selecting r items from n distinct items, even when r > n, because items can be chosen multiple times. This enables achieving a total of r selections despite having fewer unique items.

Is there a difference in how we solve combinations with repetition for large numbers?

The formula remains the same, but for large values of n and r, computational tools or software are often used to handle large factorial calculations.

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 (All on same stage)

Elements are selected and placed on a single stage, allowing them to occupy any position.
The same element can be selected multiple times since it can be repeated infinitely.
The circular movement of elements emphasises that the elements can be rearranged without affecting the outcome, highlighting that the order of selection doesn't matter.

Combination Selector

This visualisation features a hierarchical structure.
Each level of the hierarchy contains all possible combinations of the selected elements, with the number of elements matching 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 combinations.
Previous levels act as navigational tools leading to this outcome.
This design allows for easy step-by-step exploration of combinations, allowing for backtracking.