Permutations, when all objects are distinct, involve arranging unique items in a specific order without repetition. Each object maintains its individual identity, leading to various arrangements based on their distinct positions. In this article, we will discuss the case of permutation where all the objects under consideration are distinct.
Definition of Permutation
Permutation means arranging objects in a specific order. When dealing with permutations, it’s important to both choose and arrange items carefully. Basically, ordering matters a lot in permutations.
In simple terms, permutation means making an organized combination.
Types of Permutations
Permutations can be classified into three categories:
- Permutation of n different objects (without repetition): This involves arranging n distinct objects in a specific order without repeating any.
- Permutations with repetition: In this type, repetition of objects is allowed, meaning the same object can appear multiple times in the arrangement.
- Permutation when the objects are not distinct (Permutation of multisets): Here, some or all of the objects being arranged are not unique, leading to different arrangements despite having identical objects.
The formula for permutations is:
Where:
- P(n, k) represents the number of permutations of ( n ) objects taken ( k ) at a time.
- ( n! ) denotes the factorial of ( n ), which is the product of all positive integers from ( 1 ) to ( n ).
- (n – k)! is the factorial of ( n ) minus ( k ).
Permutation of n Distinct Objects
Permutation of n distinct objects refers to arranging a set of n different items in a particular order. In this type of permutation, each object is unique and cannot be repeated in the arrangement. The number of possible permutations for n distinct objects can be calculated using the formula n!, where n is the total number of objects and !! denotes factorial.
For example, if you have 3 distinct objects (A, B, and C), the number of permutations would be 3! = 3 × 2 × 1 = 6.
These permutations would include arrangements like ABC, ACB, BAC, BCA, CAB, and CBA.
Proof of Theorem: Permutation When All The Objects Are Different
To prove the theorem that deals with permutations when all the objects are different, we’ll start by defining what a permutation is. A permutation is an arrangement of objects in a specific order.
When all the objects are different, the number of permutations is calculated using factorial notation. Factorial notation is denoted by an exclamation mark (!). For example, 5 factorial (5!) is calculated as 5 × 4 × 3 × 2 × 1, which equals 120.
Now, let’s consider an example to illustrate the theorem. Suppose we have three different objects: A, B, and C. We want to find all possible permutations of these objects.
Step 1: List all the objects: A, B, C.
Step 2: Choose the first object. There are 3 choices for the first object.
Step 3: Choose the second object. After choosing the first object, there are 2 remaining choices for the second object.
Step 4: Choose the third object. After choosing the first and second objects, there is only 1 choice left for the third object.
Step 5: Multiply the number of choices at each step: 3 × 2 × 1 = 6.
Step 6: Thus, there are 6 permutations of the three different objects: ABC, ACB, BAC, BCA, CAB, CBA.
Now, let’s generalize this example. If we have n different objects, there are n choices for the first object, (n – 1) choices for the second object, (n – 2) choices for the third object, and so on until there is only 1 choice left for the last object.
Step 7: Multiply the number of choices at each step: n × (n – 1) × (n – 2) × … × 1 = n!
∴ Number of permutations when all the objects are different is n factorial (n!).
This theorem holds true for any number of different objects, as shown through the example and the generalization.
Conclusion
In conclusion, permutations are like puzzles where you arrange things in different orders. When everything is different, it’s called “permutations with distinct objects.” We’ve learned that the number of ways you can arrange these objects is calculated by multiplying their total number. So, if you have 5 different things, you can arrange them in 120 different ways!
Related Articles
Solved Examples of Permutations When All The Objects Are Distinct
Example 1: Suppose you have 5 different books, and you want to arrange them on a shelf. How many different arrangements can you make?
Solution:
Since all the books are distinct, using the formula for permutations of n distinct objects,
As we know, P(n, n) = n!
P(5, 5) = 5!
⇒ P(5, 5) = 5 × 4 × 3 × 2 × 1
⇒ P(5, 5) = 120
Therefore, there are 120 different ways to arrange the 5 books on the shelf.
Example 2: If you have 7 different colored balls and want to line them up in a row, how many different arrangements are possible?
Solution:
As we know, P(n, n) = n!
P(7, 7) = 7!
⇒ P(7, 7) = 7 × 6 × 5 × 4 × 3 × 2 × 1
⇒ P(7, 7) = 5040
Thus, there are 5040 different ways to line up the 7 colored balls.
Example 3: Consider a group of 4 friends: Alice, Bob, Carol, and Dave. They want to take a photo together, but they can arrange themselves in different orders. How many different ways can they stand for the photo?
Solution:
Since the friends are distinct individuals
With n = 4,
We have P(4, 4) = 4!/(4-4)! = 4!/0! = 4!
⇒ P(4, 4) = 4×3×2×1
⇒ P(4, 4) =24
Therefore, there are 24 different ways for Alice, Bob, Carol, and Dave to stand for the photo.
Example 4: A lock has 4 different digits (0-9) arranged in a specific order. How many unique combinations can be formed if repetition of digits is not allowed?
Solution:
Since the lock digits are distinct and repetition is not allowed, we use the permutation formula.
So, P(10,4) = 10!/(10−4)!
⇒ P(10,4) ​= 10!/6!
​⇒ P(10,4) = [10 × 9 × 8 × 7 × 6!]/6!
⇒ P(10,4) ​= 10 × 9 × 8 × 7
⇒ P(10,4) = 5,040
Therefore, there are 5,040 unique combinations for the lock.
Example 5: A password consists of 6 characters, where each character can be a lowercase letter (a-z) or a digit (0-9). How many different passwords can be created if repetition of characters is not allowed?
Solution:
Since there are 26 lowercase letters and 10 digits to choose from, the total number of characters available is 26 + 10 = 36. Using the permutation formula,
P(36,6) = 36!/(36−6)!
⇒ P(36,6) ​= 36!/30!
⇒ P(36,6) ​= [36 × 35 × 34 × 33 × 32 × 31 × 30!]/30!
⇒ P(36,6) = 1,947,792
Thus, there are 1,947,792 different passwords that can be created.
Practice Question
Q1: Arrange the letters A, B, C in all possible orders.
Q2: How many different 3-digit numbers can be formed using the digits 1, 2, and 3 without repetition?
Q3: In how many ways can a committee of 4 people be selected from a group of 7 people?
Q4: A password consists of 5 characters, where each character can be a lowercase letter (a-z) or a digit (0-9). How many different passwords can be created if repetition of characters is not allowed?
Q5: How many distinct arrangements can be made using all the letters of the word “MISSISSIPPI”?
FAQs on Permutations: When all the Objects are Distinct
What is the formula for permutations of distinct objects?
The formula for permutations of distinct objects is n!, where n represents the total number of distinct objects and !! denotes factorial.
What is the permutation when all the objects are not distinct objects?
Permutations when all objects are not distinct involve arrangements where some objects may be repeated, leading to different arrangements despite having identical objects.
What is the permutation of n distinct objects taken all at a time?
The permutation of n distinct objects taken all at a time is n!, where n represents the total number of distinct objects and ! denotes factorial.
What are permutations of distinct elements?
Permutations of distinct elements refer to arrangements where each element is unique and cannot be repeated in the arrangement.
Share your thoughts in the comments
Please Login to comment...