Permutation and Combination

Permutation and Combination both are important parts of counting. Counting the numbers with pure logic is itself a big thing. Without counting we can’t solve probability problems. This is the reason why we learn Permutations and Combinations just before probability. 

Here, we are going to see how to differentiate between Permutation and Combination, what is the difference between Combination and Permutation and the difference between Permutation and Combination with various examples.

What is Permutation?

The Permutation is a selection process in which the order matters. Permutation can simply be defined as the number of ways of arranging few or all members within a particular order. This is all about the term Permutation.

Example: The Permutations of the letters in a small set {a, b, c} are:

abc acb

bac bca

cab cba

A formula for the number of Permutations of k objects from a set or group of n. This is generally written in \[nP_k\].

Formula:

\[nP_{k} = \frac{n!}{(n - k)!} = \frac{n(n−1)(n−2)\ldots(n−n+1)}{(n-k)(n-k−1)(n-k−2)\ldots(n-k−n-k+1)} \]

There are Two Types of Permutation:

1. Permutations with Repetition

Selecting r of something(number or any element) that has n different types, the Permutations will be:

\[n \times n \times \ldots\] (r times)

(In similar words, there are no possibilities for the first selection process, THEN there are no possibilities for the second selection process, and so on, and multiplying each time.)

Usually, it becomes easy to write down using the exponent of the r:

Thus \[n^{r}= n \times n \times \ldots\] (upto r times)

So, the general formula is simply: \[n^{r}\]

where n is the number of elements to choose from (ie. set or sink of elements) and we choose r of them, repetition is allowed, and order matters.

2. Permutations Without Repetition 

Without repetition, our choices get reduced each time.

Let’s take the kind of most easy and widely used example:                    

How many different 4-card hands can be made from a deck of cards?

In this problem, the order is immaterial since it doesn’t matter what order we select the cards. We begin with four lines to represent our 4-card hand

Assuming all the 52 cards available for the first draw, place “52” in the first blank. Once you choose a card, means one card is already selected so there will be one less card available on the next selection draw. So the second blank there will be 51 options available. Also, The next draw will have two fewer cards in the deck, so there are now 50 options, and so on. The formula is written:

\[P\binom{n}{r} = nP_{r} = \frac{n!}{(n - k)!} \]

Using the formula we get

\[P\binom{52}{4} = 52P_{4} = \frac{52!}{48!} \]

where n is the number of things to choose from (ie. set or sink of elements), and we choose r of them, no repetitions and order matters.

What is Combination?

Combination is the way of selecting items from a bulk collection, such that (non-similar Permutations) the order of selection does not matter. We can say in smaller cases, we will be able to count the number of Combinations. Combination refers to the Combination of n things taken k at a time without repetitions. A Combination is the choice of r things from a set of n things without any replacement and where order doesn't matter.

\[C\binom{n}{r} = nC_{r} =\frac{nP_r}{r!} = \frac{n!}{r!(n - k)!} \]

Let’s take an example and understand this,

We have three digits (1,2,3) and we want to make a three-digit number, So the following numbers that will be possible are 123, 132, 213, 231, 312, 321..

Combinations give us an easy way to work out how many ways "1 2 3" could be placed in a particular order, and we have already seen it. The answer is:

3! = 3 \[\times\] 2 \[\times\] 1 = 6

So we reprint our Permutation’s formula to reduce it by how many ways the objects could be in order (because we aren't interested in their order any more).

Difference between Permutation and Combination with Examples

It is neither too easy nor too difficult to get the Permutation and Combination difference. We’ll see some examples to understand the difference between them.

Permutations

• Arrangement of people, digits, numbers, alphabets, letters, and colours etc.

• Picking a team captain or keeper and a particular one from a group.

• Picking two favourite colours, in order, from a colour book.

• Picking first, second and third prize winners.

Combinations

• Selections of the menu, food, clothes, subjects, team etc.

• Picking three team members from a group.

• Picking two colours from a colour book.

• Picking three winners only.

How to Differentiate Between Permutation and Combination

Permutations and Combinations, refers to the various ways in which objects from a set may be selected, generally without replacement, to form subsets (or we can say the number of subsets for a set). This selection of subsets is called a Permutation when the order of selection is a factor, a Combination when order is not a factor. (In simple words selection of subsets is a Permutation and the non-fraction order of selection is called Combination).

Similarities Between Permutation and Combination

In terms of mathematical concepts, “Permutation” and “Combination” are related to each other. Combination is the counting of selections that we make from n objects. Whereas Permutation is counting the number of arrangements from n objects.

The point we need to keep in our mind is that Combinations do not place an emphasis on order, placement, or arrangement but on choice.

How can students revise for Permutations and Combinations on Vedantu?

Vedantu is a reliable online tutoring platform for students and can be used by all students absolutely free of cost. It has relevant material on Permutations and Combinations for studying if one goes to Know about the difference between Permutations and Combinations. 

This page has defined the basics of what each is and then it goes on to describe the similarities and the differences. Everything has been explained in elaborate and simple language. How the selection of music, food, clothes and other everyday items takes place has been explained.

Permutations and Combinations is a pretty interesting topic that needs to be dealt with in a strategic manner.

Where will students find out about the difference between Permutations and Combinations online?

Students can look for the same on Vedantu. This page is very informative in terms of explaining Permutations and Combinations. This chapter is crucial in Maths and if the students grasp the basics of this topic, they will be well prepared for topics like Probability and Statistics later on. Students just need to log in into Vedantu’s portal so as to get access to these.