If there is one topic in competitive exams, which almost everyone struggles with, it is Permutation & Combination. Almost all the papers have 2-4 questions from this topic and almost everyone leaves these questions. However, these are very easy and take less than a minute to solve. Only thing you need to do is understand the concept.
Combination means selecting things at random (order doesn't matters) while Permutation means arrangement of objects in particular order.
For example, from all the subjects and sections that can be asked in an exam, you have to study combination of 3 sections for a bank exam: Quant, Reasoning and Verbal. Here it doesn't make any difference if you write the above sections as reasoning, verbal and quant. The sections to attempt will remain same.
The order, in which you will attempt the section, however will be different. Some people will attempt in the order of verbal, quant and reasoning. Some others may want to attempt in the order - verbal, reasoning and quant. Here the order does matter. So the total ways in which someone can attempt the complete paper (3x2x1 = 6) is known as permutation.
A simpler way to remember is:
Permutation sounds complex. Well it is because in permutation each and every detail matters.
Combination is simpler and easy: Details don't matter.
Permutation is a tool to make lists where order matters whereas combination is used for groups where order doesn't matters.
With the difference clear between the two, it is time to go deeper into the ocean of permutations and combinations.
Let n be a positive integer. Then n factorial (n!) can be defined as
n! = n(n-1)(n-2)...1
5! = 5 x 4 x 3 x 2 x 1
0! = 1
1! = 1
Suppose, there is a swimming competition going on and we have 8 participants: A, B, C, D, E, F, G and H.
We have to award Gold, Silver and Bronze medals to the top 3 contestants respectively. In how many ways can award the medals to the 8 participants.
We will use permutations since the order in which we have to distribute the medals matters.
- To give away the gold medal we have: 8 choices A B C D E F G H. Assume A wins the Gold.
- Now for the Silver medal we are left with 7 choices (since A already won the gold medal): B C D E F G H. Assume B wins the silver.
- With A and B already bagging gold and silver we are left with 6 choices for the Bronze medal: C D E F G H. Let's assume C wins the bronze.
Note we had 8 choices at first, then 7, then 6. Hence, the total number of ways was 8 * 7 * 6 = 336.
Digging deeper into the details, we have to arrange 3 people out of 8 competitors. To achieve this, we started with all the options (8) then took them away one at a time (7, then 6) until there were no medals left.
A factorial of a number is obtained by multiplying all the natural numbers starting from 1 to the number.
So factorial 8 will be 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1( Factorial of a number n is denoted by n!)
But we need only 8 x 7 x 6 for our answer.
We need to remove 5 x 4 x 3 x 2 x 1. Notice it is the expression for 5!.
So, if we do 8!/5! we get: 8 x 7 x 6 (remaining gets cancelled as per division law)
You must be wondering why we took the number 5? Because, 5 people were left after we awarded 3 medals from 8.
i.e. 8!/(8-3)! "Use the first 3 out of 8!".
So, If we have n items total and want to pick k in a particular order, we get: n!/(n-k)!
This is the permutation formula: From a total of n items, you need to find number of ways in which k items can be ordered:
We hope that now you have much better knowledge about the difference between Permutation and Combination, as well as basic concepts of Permutation. In second part of the series, we will discuss about Combination.
The article is written by Vaibhav Mehta, Education Consultant at www.MockBank.com, a Bengaluru based online Test Preparation Company for government/PSU jobs