Permutation vs. Combination – How Different are They?

Permutation and combination are quintessential parts of a high-school math curriculum. While the chapter may seem comparatively easier than other math chapters for permission to use the calculator, it isn’t easy all the way. Unless you understand which problem can be solved via permutation and which by combination, the chances are that you might lose some valuable marks. So, how different are these two concepts?

Permutation vs. Combination – The Basics

The fundamental difference between permutation and combination is determining the order of the objects or variables in question. In permutation, you need to focus on the arrangement of the number of objects and understand which variables are taken few times and which all at a time. In contrast, the order does not matter when it comes to solving a probability issue using combination.

Permutation vs. Combination

Apart from mathematics, permutation and combination are used in practical life too. As shocking as it may be, poets use permutation to decide the syllables in a line of a verse. Again, the schedules for sports matches are determined using permutation too. Combinatorics is also used by businesses to make production-related decisions. Read on to know more about permutation vs. combination from the comparison chart given below.

Permutation vs. Combination: Comparison Chart

Factor PERMUTATION COMBINATION
Meaning Permutation can be defined as the different ways how we can arrange a set of objects in a sequence. Combination means the different ways of choosing variables or items from a set of objects not focusing on the order of the same.
Order The focus is on the order of the positioning of the variables or items Order is irrelevant in combinatorics.
Denotes The arrangement of variables The selection of items in no particular order
What is it? Set of elements in a logical sequence Sets of items without any order
Answers Permutation tells us how many groups can be created from a set of objects. Combinatorics tells us how many different groups can be selected from a larger group of items.
Derivation Multiple permutations from a single combination A single combination from a single permutation

Permutation: Definition, Example, and Formula

Permutation is defined as the different ways how all members in a set can be arranged in a specific order. Permutation helps us arrive at the possible arrangement or rearrangement of a set in a distinguishable order.

Let us understand with an example.

 Below, we are going to create possible permutations with the letters A, B, and C–

  • Taking two letters at a time, we get – ab, ac, ba, bc, ca, cb.
  • Taking all the three letters at a time, we get – abc, acb, bac, bca, cab, cba.

Formula to determine permutation:

The formula to calculate the total number of possible permutations of n number of things and taken r at a time is:

Combination: Definition, Example, and Formula

Combinatorics is all about determining the different ways of creating a group by choosing a few of the items of a set or the entirety in no particular order.

Let us understand with an example.

 Let us create every possible combination with the letters X, Y, and Z.

  • Taking all three out of three letters, the only combination is xyz.
  • Taking two out of three letters, the possible combinations are xy, yz, zx.

The formula to calculate the total number of possible combinations of n number of items and taken r at a time is:

Permutation vs. Combination: Key Differences

Permutation and combination may often seem similar but are two very different concepts. Understand the differences between permutation and combination based on the following grounds:

  1. Permutation is all about arranging a set of items in several ways while following a sequential order such that no two items are set in a repeated order. In combination, you have to choose items from a large set and determine how to create sub-sets with no relevant order.
  2. Order, placement, and position are the factors that play the most important role when it comes to distinguishing the characteristics of permutation from that of combination.
  3. Permutation is used to arrange things, digits, alphabets, people, colours, etc. For example, it is used to determine sports schedules, phone numbers, and seating arrangements. On the other hand, we use combinations on a daily basis while selecting delicacies from a menu, mixing and matching our clothes, etc.
  4. We can derive many permutations from a single combination. On the contrary, we can derive just one combination from a single permutation.
  5. Permutation can solve the problem of creating different arrangements for items from a set. In comparison, combination explains the number of groups that can be formed from a larger set of objects.

Let us now understand the application of permutation vs. combination using one example.

If you are asked to find out the total possible sets created with two out of three objects (A, B, and C), you must first understand whether you will have to apply permutation or combination. To find that out, try to figure out whether order plays an important role or not.

If order is crucial, it is about permutation. Then the possible samples will be AB, BA, BC, CB, AC, CA. Note how each subset is different from the other.

If order is not the focus, you have to apply combinatorics. In that case, the possible samples should be AB, BC, and CA.

Parting thoughts,

We hope that this comprehensive guide with examples could help you understand the basics of permutation vs. combination. Bookmark this post and go through it when you cannot understand which to apply. Since permutation and combinatorics are used in mathematics, statistics, and research, it is crucial to get the difference between the two clearly.

If you still cannot figure out the difference between the two, take your time and use resources beyond your textbooks. You can hire our math homework expert math and statistics experts or pore through our archive of samples on this particular topic.

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