Combination as section

Introduction

A combination is a mathematical approach that we study to find the number of possible arrangements in the collection of objects where the order of the selection is not defined. In combinations, the items can be selected in any order. Apart from mathematics, this chapter also has numerous uses in real life. This topic is one of the most integrated topics of mathematics as many mathematical formulas and topics like permutation and combination, sets, combination as section, etc are based on the concept of combination. This topic is very important for the IIT JEE Examination.

What are Combinations?

A combination is a mathematical algorithm used to calculate the possible number of arrangements of a group of objects where the selection order does not matter. You can select the objects in any order. 

In simple terms, we can say that Combinations are a selection process. The major difference between Combinations and permutations is that in permutation the order of selection is important. For example, the arrangements xy and yx are equivalent in combinations, but in permutations, the arrangements vary. 

For example, if we want to purchase  an ice cream and you are allowed to combine any three flavors from vanilla, chocolate, red cherry, and butterscotch, then the combination of vanilla, chocolate, and red cherry is the same as the combination of chocolate, red cherry, and butterscotch. So, let us shorten the name of the fruits by selecting the first letter of their names. We have four expected combinations for the question above ABC, ABD, ACD, and BCD. 

What is the basic principle of counting?

The fundamental principle of counting is a mathematical algorithm used to calculate the total no. of possible results in a condition. This formula denotes that if there are p possible ways of doing work and q possible ways of doing another work, then there are total p x q options available to perform the task. 

Combinations Formula

As we have read above, Combination basically means, “arrangement and selection of the object or a group of objects in different ways”.

The mathematical formula for combination is:-

Using permutation rule, combination formula is:-

In the above formulas, notations are as follows:-

• r denotes the size of the permutation
• n denotes the size of the set 
• n, r are positive integers (non-negative integers)
• ! is the factorial symbol

There are two fundamental principles of counting on which this topic is based. The first principle is the fundamental addition principle counting and the fundamental multiplication principle of counting.

Let us say we have 3 pens and 2 markers. Every item is distinct and no 2 items are the same. Here’s your first question:

(i)   In how many ways can we pick just one of each item?

We can either pick the first pen or can pick the second pen or maybe the third pen or maybe we pick one of the markers. So, we have a total five-way to choose any of the items. Therefore, in each case, we have different possibilities, and if we add all the possibilities, we will get 5 options. So, this is called the additional principle of counting.

(ii) In how many ways, can we pick one pen and one marker?

We can pick one of the 3 pens and then we need to pick one of the 2  markers. Maybe after picking the first pen, we pick the first marker, or maybe after picking the second marker we can pick the first pen. So, one way is P1 M1, and another is P1 M2, or maybe we pick P2 and then M1 or P2 and M2. So, in this case, we have 6 ways to pick a pen or a marker.

Which are the six ways?

P1 M1, P1 M2, P2 M1, P2 M2, P3 M1, P3 M2

It is clear that we get 3×2=6 ways of choosing items. So, another rule of counting is the multiplication rule.

Mathematical expression: Permutation and Combinations 

There are several formulas used in permutation and combination concepts. The 2 main Permutation and Combinations Formulas are as follows:-

Permutation Formula

nPr = (n!) / (n-r)!

Combination Formula

nCr= n/r = nPr/r! = n!/r! (n – r)!

Difference Between Permutation and Combination

Permutation refers to the different ways of arranging a set of objects in a sequential order. Combination refers to several ways of choosing items from a large set of objects, such that their order does not matter.

What are the uses of Permutation and Combination in real life?

You can use the knowledge of permutation and combination in day-to-day life activities. 

You have 3 Shirts and 2 Pants. Using these shirts and pants, you want to create a maximum no. of outfits of pants and shirt.

You want to paint your home. Your room has 3 different walls and you have 3 different colors to paint. You can use your knowledge of combinations to color the walls in different ways.

There are numerous daily life examples available in which you can use these concepts to make your daily life much easier.

Conclusion

It’s more important to understand why permutations and combinations work than it is to memorize the formulas. If the order doesn’t matter, it is a combination. If the order does matter it is a permutation.