Combination Formula: Definition, Relation, Difference, Examples

In mathematics, a Combination Formula represents a method of choosing items from a collection, where the specific order of selection is not considered. To illustrate this concept, consider a set containing three numbers: P, Q, and R. Determining how many ways we can choose two numbers from this set is precisely what combinations are all about.

In simpler scenarios, counting the number of combinations can be straightforward. However, when dealing with larger sets or groups of elements, the potential combinations can quickly become numerous. Therefore, a formula has been developed to calculate the possible ways of selecting a certain number of items, which we will explore in this article. Additionally, we will delve into the relationship between permutation and combination, supported by theorems and their corresponding proofs.

Definition of Combination in Math

In mathematics, a combination Formula is a selection of items from a larger collection or set, where the specific order of the selected items is not considered. Combinations are used to calculate the number of ways you can choose a subset of items without regard to the arrangement or sequence in which they are chosen.

The term "combination" is often denoted as "C(n, r)" or "n choose r," where "n" represents the total number of items in the set, and "r" represents the number of items you want to choose from the set. The formula for calculating combinations is:

C(n, r) = n! / (r! * (n - r)!)

In this formula, "n!" (read as "n factorial") represents the factorial of "n," which is the product of all positive integers from 1 to "n." The combination formula calculates the total number of ways to select "r" items from a set of "n" items without considering the order in which they are chosen.

Combinations are commonly used in various mathematical and practical applications, including probability theory, statistics, and combinatorics, to solve problems involving selecting items from a larger group without arranging them in a particular sequence.

Combination Formula

The number of ways to select 3 objects from a set of 4 objects, which is denoted as "4C3," is equivalent to the number of possible subgroups of 3 objects that can be chosen from the same set of 4 objects. For example, if you have three fruits: an apple, an orange, and a pear, there are three combinations of two fruits that can be selected from this set: an apple and a pear, an apple and an orange, or a pear and an orange.

To define this concept more formally, a k-combination of a set represents a subset consisting of k distinct elements from the set. If the set contains n elements, the count of k-combinations can be expressed using the binomial coefficient formula:

nCk = [(n)(n-1)(n-2)….(n-k+1)]/[(k-1)(k-2)…….(1)]

which can be written as;

nCk = n!/k!(n-k)!,  when n>k

nCk = 0,  when n<k

Where n = distinct object to choose from

C = Combination

K = spaces to fill (Where k can be replaced by r also)

The combination can also be represented as nCr, nCr, C(n,r), Cr

Also Check – Quadratic Equations Formula

Relation Between Permutation and Combination

Combination and permutation are related concepts in combinatorics, but they differ in how they treat the order of selection. Combination is a selection where the order is not considered, whereas permutation considers the order of selection. Consequently, the count of permutations is always greater than the count of combinations, which is a fundamental distinction between the two.

Now, let's explore the relationship between permutation and combination through the following theorem:

Theorem: nPr = nCr * r!

According to this theorem, for every combination represented by nCr, there exist r! permutations. This is because, within each combination, the r-selected objects can be rearranged in r! different ways. This theorem establishes a clear connection between permutation and combination, highlighting the relationship between the two concepts.

Also Check – Probability Formula

Difference Between Permutation and Combination

Definition:

Permutation: Permutation refers to the arrangement of objects in a specific order. It involves selecting and arranging items from a set in a particular sequence.

Combination: Combination, on the other hand, involves selecting items from a set without regard to their order. It focuses on choosing items without arranging them in a specific sequence.

Order Matters:

Permutation: Order matters in permutation. Changing the order of selected items results in a different permutation. For example, arranging letters of the word "ABC" in different orders like ABC, BCA, CAB, etc., are distinct permutations.

Combination: Order does not matter in combination. Selecting the same items in a different order does not change the combination. For instance, choosing three people from a group to form a committee results in the same combination, regardless of the order in which they were chosen.

Formula:

Permutation: The formula for permutations is P(n, r) = n! / (n - r)! where 'n' is the total number of items, 'r' is the number of items to be selected, and '!' denotes factorial.

Combination: The formula for combinations is C(n, r) = n! / (r! (n - r)!), where 'n' is the total number of items, 'r' is the number of items to be selected, and '!' represents factorial.

Examples :

Permutation: Finding the number of ways to arrange five books on a shelf, choosing the order of runners in a race, or arranging seats for a group of people at a round table.

Combination: Determining the number of ways to select a group of three students to form a study group, choosing a committee from a pool of candidates, or picking ingredients for a recipe.

Distinctness:

Permutation: Permutations are distinct arrangements, so changing the order changes the permutation.

Combination: Combinations are not distinct arrangements, and the same combination can be achieved in multiple ways by selecting the same items.

In summary, the fundamental difference between permutation and combination lies in whether order matters. Permutation focuses on arranging items in a specific sequence, while combination deals with selecting items without regard to their order. Both concepts are essential in solving various combinatorial problems.

Also Check – Conic Section Formula

Solved Examples

Example 1: Consider a group of 3 lawn tennis players: S, T, and U. You need to form a team consisting of 2 players. How many ways can you do this?

Solution: In a combination problem, the order of arrangement or selection is irrelevant. Therefore, ST (S and T as a team) is the same as TS (T and S as a team), and similarly for the other combinations.

So, there are 3 ways to select a team: ST, TU, and SU.

Using the combination formula, we have:

3C2 = 3! / (2! * (3 - 2)!)

= (3 * 2 * 1) / (2 * 1 * 1) = 3

Therefore, there are 3 different ways to form a team of 2 players from the group.

Example 2: Find the number of subsets of the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} that contain 3 elements each.

Solution: The given set contains 10 elements, and you want to create subsets with 3 elements each, regardless of the order. This is a combination problem, and we can use the combination formula.

So, the number of subsets with 3 elements is:

10C3 = 10! / [(10 - 3)! * 3!]

= (10 * 9 * 8) / [(3 * 2 * 1)]

= 120 ways.

Therefore, there are 120 different subsets of 3 elements that can be formed from the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.