What is Combination in Math Formula with Example

Combinations are an important concept in mathematics that deals with the selection of items from a given set. It is a branch of combinatory, which is a field of mathematics that studies the ways in which objects can be arranged or combined. Combinations are useful tools in solving many problems in different fields, such as statistics, probability, and computer science.

In this article, we will explore what combinations are in math formulas and provide examples to help you understand the concept.

 Definition of Combinations

Combinations refer to the selection of objects from a given set where the order does not matter. In other words, combinations are arrangements of a set of objects in which the order is not important.

For example, if you have five different fruits, say an apple, a banana, an orange, a mango, and a pear, and you want to select any three of them, the selection will be a combination, and the order in which you select them will not matter. The number of possible combinations that can be made from a set is determined by a formula.

Permutation vs. Combination

Permutations are different from combinations, as the order matters in permutations. For instance, if you have the same set of five fruits and want to select any three, the selection will be a permutation, and the order in which you select them will matter. The formula for permutations is different from that for combinations.

Formula for Combinations

The formula for combinations is expressed as ⁿC_k, where n is the total number of objects in the set, and r is the number of objects selected. The formula is given as:

ⁿC_r = n! / (r! * (n-r)!)

Where! denotes factorial.

For example, if you have six books and you want to select any three of them, the number of possible combinations is:

6C3 = 6! / (3! * (6-3)!) = 20

Therefore, there are 20 possible combinations of three books that can be selected from a set of six.

Example 1: Selecting a Committee

Suppose there are ten people in a group, and you want to select a committee of four people. How many possible committees can be formed?

Solution:

n = 10, r = 4

¹⁰C₄ = 10! / (4! * (10-4)!) = 210

There are 210 possible committees that can be formed.

Example 2: Choosing a Combination Lock

A combination lock has three dials, each numbered from 0 to 9. How many possible combinations are there?

Solution:

n = 10, r = 3

¹⁰C₃ = 10! / (3! * (10-3)!) = 120

There are 120 possible combinations that can be set on the lock.

Example 3: Distributing Party Favors

Suppose you have eight different party favors, and you want to distribute them among four guests. How many possible ways are there to distribute the party favors?

Solution:

n = 8, r = 4

⁸C₄ = 8! / (4! * (8-4)!) = 70

 Proof of the Formula for Combinations

The formula for combinations can be derived using the principles of counting and factorials. Suppose we have a set of n distinct objects, and we want to choose r objects from this set without considering the order of selection. To find the number of possible combinations, we can use the following approach:

Choose the first object in n ways.

Choose the second object in (n-1) ways, as one object has already been selected.

Choose the third object in (n-2) ways, as two objects have already been selected.

Continue this process until r objects have been selected.

Using the multiplication principle, the total number of ways to choose k objects from a set of n objects is:

n * (n-1) * (n-2) * … * (n-r+1)

This can be simplified as follows:

n * (n-1) * (n-2) * … * (n-r+1) / r!

Using the definition of factorials, this can be further simplified as:

n! / (r! * (n-r)!)

Therefore, the formula for combinations is ⁿC_r = n! / (r! * (n-k)!), which gives the number of possible ways to choose k objects from a set of n objects without considering the order of selection.

Key Takeaways

Combinations are arrangements of a set of objects in which the order is not important.

The formula for combinations is ⁿC_r = n! / (r! * (n-r)!)

Permutations are different from combinations, as the order matters in permutations.

Combinations are useful in solving problems in different fields, such as statistics, probability, and computer science.