Permutations

A permutation is an arrangement in which the order of the items matters.
Consider an example in which five chairs are available, and three people need to be seated.
 

• There are five choices for the first chair.
   
• After seating one person, four choices remain for the second chair.
   
• Then, three choices are left for the third chair.
   

So, the total number of ways to arrange three people in 5 chairs is:

5 × 4 × 3 = 60 ways

Notice that this multiplication can be written using factorials:

\(5 \times 4 \times 3 = \frac{5!}{2!} = \frac{5!}{(5-3)!}\)

In general, when arranging r people in n chairs, the number of permutations is given by:
 
\(nP_r = \frac{n!}{(n-r)!}\)
 

Here, let’s see how we can find permutations. The general formula we use to find permutations is:

P(n, r) = n! / (n − r)!

Here, n is the total number of elements in the data set.

r is the total number of selected elements in a specific order.

! is the factorial. 

For instance, let's say we have 10 different books and want to select and arrange 2 of them. We can calculate the number of ways to do it using permutations. 

Without repetition:

The formula is: ⁿP_r = 10! / (10 − 2)! = 10! / 8! = (10 × 9 × 8!) / 8!  = 90

So, there are 90 unique ways to arrange 2 books from a set of 10. 

With repetition, the formula is:

10\(^2\) = 10 × 10 = 100

Hence, there are 100 ways (with repetition) to arrange 2 books from a set of 10. 

Another key concept is factorials, and they are useful in permutations.

For example, the factorial of 8 = 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40,320.

Permutations has a different form depending on the rules for arranging the object. To understand how arrangements work in various situations, permutations are classified into three main types. Each type follows a different condition about repetition and the uniqueness of objects.

• Permutation of n different objects (without repetition)
   
• Permutation with repetition
   
• Permutation of multisets (when some objects are identical)

1. Permutation Without Repetition

All objects are different, and no object is used more than once. Each time you choose an object, the number of choices decreases.

The formula is:

\(P(n, r) = n \times (n - 1) \times (n - 2) \times (n - r + 1)\)

This means we are arranging r objects from n distinct objects, and order matters.
 

2. Permutation With Repetition

Objects can be used again and again. Since repetition is allowed, the number of choices stays the same each time.

The formula is:

\(P_{\text{repetition}}(n, r) = n^r\)

This means each of the r positions can be filled in n ways.

3. Permutation of Multisets

In this type, some objects are repeated or identical. Since some items look the same, we adjust the count to avoid repeating the same arrangement.

A multiset has groups of identical items, and the formula ensures only unique arrangements are counted.
 

To understand permutations better, it helps to know a few important properties. These properties show how permutation values behave and how they can be simplified or related to each other.
 

Here are the commonly used properties of permutations:

• \(nP_n = n \times (n-1) \times (n-2) \times \dots \times 1 = n!\)

   

• \(nP_0 = \frac{n!}{(n-0)!} = \frac{n!}{n!} = 1\)

   

• \(nP_1 = n\)

   

• \(nP_{n-1} = \frac{n!}{(n-(n-1))!} = \frac{n!}{1!} = n!\)

   

• \(\frac{nP_r}{nP_{r-1}} = n - r + 1\)

   

• \(nP_r = n \cdot (n-1)P_{r-1} = n \cdot (n-1) \cdot (n-2)P_{r-2} = n \cdot (n-1) \cdot (n-2) \cdot (n-3)P_{r-3} \dots \), and so on.
   
• \((n-1)P_r + r \times (n-1)P_{r-1} = nP_r\)

Permutations and combinations are methods used to determine the number of possible arrangements of elements. So, let's see how they differ from each other. 

Understanding permutations that become easier when the ideas are explained using the  simple language and real-life examples. These tips help make the concept clearer, more interactive, and easier to apply in different situations.
 

• If you change the order, the arrangement becomes different.
   
• Use small examples first to build understanding before solving larger problems.
   
• Connect the concept to real-life examples, such as arranging books, toys, or snacks in different orders.
   
• Use a permutation calculator when numbers become large to help children verify answers.
   
• Explain permutation vs combination through simple scenarios: arranging students (permutation) vs choosing a group (combination).
   
• Start with practical situations like arranging students in a line or organizing items on a shelf.
   
• Simplify the process by multiplying just the first r terms rather than writing out the whole factorial.
   

Now let’s learn how we use permutations in the real world.

 

• In cybersecurity and encryption, we use permutations to create passwords to secure our systems. 
  
   
• In lottery games, we use permutations to pick from a pool of numbers. 
  
   
• Permutations can be applied to various scenarios, such as arranging people, and seats, forming teams, and so on.
  
   
• In DNA and genetics, permutations help in analyzing sequences of a DNA.
   
• Permutations are used to organize schedules, plan tasks, or determine the order of events efficiently.
   
• Permutations are used to determine the possible orders of winners, medal distributions, and match schedules.