Combination Calculator

Type:

Total Elements (n)

Selected Elements (r)

ⁿC_r

⁴C₃

Total Elements (Individuals) [4] 𝒊

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

ABC

Combination Selector 𝒊

➔

ABC[#1]

ABD[#2]

ACD[#3]

BCD[#4]

Combination

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 a standard combination, each element can only be selected once, and the number of ways to choose a group of elements from a larger collection is based on the available choices. Combinations are commonly used in probability, statistics, and various real-life scenarios where the arrangement of items is irrelevant but the selection matters.

Combination Formula

In cases where we want to select elements from a group without regard to the order, we can determine the number of possible combinations using the combination formula:

^{n} C_{r} = \frac{n!}{r! \cdot (n - r)!}

nCr = Combination of distinct elements taken at a time | n = total number of elements | r = number of elements to choose

Combination Examples

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

Problem: How many ways can 3 students be chosen from a group of 5 students?
Solution: Using combination formula: 5! / [3! × (5 - 3)!] = 10.
Answer: There are 10 ways to choose the students.

Example 2: Combinations of Fruits

Problem: How many ways can 2 fruits be selected from a basket of 6 different fruits?
Solution: Using combination formula: 6! / [2! × (6 - 2)!] = 15.
Answer: There are 15 ways to select the fruits.

Example 3: Combinations of Cards

Problem: How many ways can 5 cards be chosen from a deck of 52 cards?
Solution: Using combination formula: 52! / [5! × (52 - 5)!] = 2598960.
Answer: There are 2598960 ways to choose the cards.

Combination Exercise

Engage in this Combination 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 4 students be chosen from a group of 6 students?
Ans 1: 15.
Que 2: How many ways can a committee of 3 members be selected from 8 people?
Ans 2: 56.
Que 3: How many ways can 4 fruits be selected from a basket of 7 different fruits?
Ans 3: 35.
Que 4: How many ways can 6 cards be chosen from a deck of 52 cards?
Ans 4: 20358520.
Que 5: How many ways can a team of 2 players be formed from 5 available players?
Ans 5: 10.

Combination Calculator FAQ

How is a combination different from a permutation?

In a combination, the order of items does not matter, whereas, in a permutation, the order does matter. For instance, choosing 3 students from 5 is a combination, but arranging 3 students in a line from 5 is a permutation.

How do you use combinations in probability?

In probability, combinations are used to calculate the likelihood of certain outcomes by determining the number of favourable outcomes and the total number of possible outcomes. For example, calculating the probability of drawing a certain hand in poker involves combinations.

How is the combination C(n, 0) equal to 1, and what does it mean?

C(n, 0) represents the number of ways to choose 0 items from a set of n items. It equals 1 because there is exactly one way to choose nothing from a set: by choosing none at all. This means that regardless of the number of items in the set (as long as n is non-negative), there is always one way to select none.

Total Elements (Individuals)

This visualisation represents the collection of distinct elements.
Each element is unique, and helps in generating combinations.

Selected Elements (All on same stage)

Elements are selected and placed on a single stage, allowing them to occupy any position.
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.