Combination and permutation both play an important role in counting. To count in a logical way is an important thing. It is impossible to solve problems involving probability without counting. Permutations and combinations are learned just before probability because of this reason.
With various examples, we will learn how to distinguish between permutation vs combination, what is the difference between combinations and permutations, and how permutations and combinations are different.
What is Permutation?
The order of permutations matters in this selection process. You can define permutation as the number of possible arrangements of a few or all members within a particular order. That's all there is to it.
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 nPk.
Formula:
nPk=
n!
(n−k)! =n(n−1)(n−2)....(n−k+1)
knP=n!(n−k)!=n(n−1)(n−2)....(n−k+1)
There are Two Types of Permutation:
1. Permutations with Repetition
The permutations are as follows when selecting r of something with n different types:
n × n × ... (r times
The first selection process has no possibilities, THEN there are no possibilities for the second selection process, and so on, and so forth.
By using the exponent of r, it becomes easier to write down:
Thus nr=n × n × ... (up to r times)
In general, the formula is:
nr
We can choose r of the available elements (sink or set of elements), and repetition is permitted and order matters.
2. Permutations Without Repetition
Whenever we do not repeat something, our choices become fewer.
As an example, let's use one of the most common and easiest ones:
What is the maximum number of different four-card hands possible from a deck of cards?
It doesn't matter what order we select the cards in since the order is immaterial in this problem. To begin, we draw four lines to represent our four cards.
For the first draw, assume that all 52 cards are available. Write "52" in the first blank. After you choose a card, the next selection draw will have one less card, since one was already selected. That leaves 51 options available for the next blank. In addition, there will be two fewer cards in the deck for the next draw. This means that there will be 50 options. Here is the formulaR(n,n) = P(n,n):
P(n,r)=nr
P=n!(n−r)!
P(n,r)=rnP=n!(n−r)!
Using the formula we get
P(52,4)=
52
4
P=52!
(48)!
P(52,4)=452P=52!(48)!
n is the number of elements we need to choose from (i.e. the sink or set of elements to choose from), and r is the number of elements to choose from; the order matters.
What is Combination?
In combination, items are selected from a collection in such a way that it does not matter the order in which they are selected (non-similar permutations). If the cases are smaller, we can count the combination numbers. As a rule, a combination consists of n things taken k at a time without repetition. In a combination, r things are chosen without replacement from a set of n things and the order does not matter.
Let’s take an example and understand this,
The numbers that are possible are 123, 132, 213, 231, 312, 321. We are given three digits (1,2,3) and want to make a three-digit number.
By using combinations, we can easily figure out how many ways "1 2 3" can be arranged in a particular order. Here is the answer:
3! = 3 × 2 × 1 = 6
Therefore, we reprint our permutation's formula to reduce it by how many ways the objects can be arranged (since we're not interested in their order any more).
Difference between Permutation and Combination with Examples
Permutation and combination differences are neither too easy nor too difficult to calculate. Here are a few examples to help you understand the difference.
Permutations
People, numbers, letters, alphabets, and colors are arranged.
Choosing a team captain or goalkeeper from a group.
Picking two of your favorite colors, in order, from a color book.
Choosing the winners of the first, second, and third prizes.
Combinations
Food and clothing selections, subject selections, team selections, etc.
Selecting three team members from a group.
A colour book is used to pick two colours.
Only selecting three winners.
How to Differentiate Between Permutation and Combination
Combination and permutation refers to the various ways in which objects from a set can be selected without replacement to form subsets (or the number of subsets within a set). When the order of selection is a factor, this selection of subsets is called a permutation, while when it is not a factor, it is called a combination. (Simply put, selecting subsets is a permutation, while the non-fraction order of selection is a combination.)
Similarities Between Permutation and Combination
The mathematical concept of "permutation" and the concept of "combination" are related. From n objects, we make combinations by counting the selections we make. By contrast, permutation refers to the number of arrangements that can be derived from n objects.
Keeping in mind that combinations do not place an emphasis on order, placement, or arrangement but rather on choice, is very important.
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