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.

Formula of Permuation

The formula for permutations is:

[Tex]\bold{P(n, k) = \frac{{n!}}{{(n – k)!}}}[/Tex]

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