Understanding Permutation and Combination

Permutation and combination are fundamental concepts in statistics and mathematics that are used to determine the total number of ways to arrange or select a specific set of objects. In this article, we will provide a comprehensive understanding of permutation and combination, and how they differ from each other.

Introduction

Permutation and combination are closely related to each other, but they differ based on whether the order matters or not. Permutation refers to the total number of possible arrangements of a set of objects, where the order matters. Combination, on the other hand, refers to the total number of possible selections of a set of objects, where the order doesn’t matter.

Permutation

Permutation deals with the number of ways to select objects in a particular order. In other words, it calculates the total number of possible arrangements of a set of objects, where the order of the objects matters.

For example, consider the word “ABC”. There are six possible permutations of the letters “A”, “B”, and “C”, as follows:

  • ABC
  • ACB
  • BAC
  • BCA
  • CAB
  • CBA

The total number of possible permutations of n objects taken r at a time is given by the formula:

nPr = n! / (n - r)!

Here, n! refers to the factorial of n, which is the product of all positive integers from 1 to n. Therefore, the formula for permutation is the number of permutations of n objects taken r at a time.

Combination

Combination deals with the number of ways to select objects without considering their order. In other words, it calculates the total number of possible selections of a set of objects, where the order of the objects doesn’t matter.

For example, consider the word “ABC” again. There are only three possible combinations of the letters “A”, “B”, and “C”, as follows:

  • ABC
  • ACB
  • BAC

Notice that these are the same three permutations listed above, but they are considered as a single combination because their order doesn’t matter.

The total number of possible combinations of n objects taken r at a time is given by the formula:

nCr = n! / (r! * (n - r)!)

Here, r! is the factorial of r. Therefore, the formula for combination is the number of combinations of n objects taken r at a time.

Example

Let’s take another example to understand the difference between permutation and combination. Suppose we have a set of four numbers: {1, 2, 3, 4}.

Using Permutation: The total number of possible permutations of two numbers from this set is given by:

4P2 = 4! / (4 - 2)! = 4! / 2! = 12

Here, the order matters, so we need to consider all possible arrangements of two numbers from the set. Therefore, the total number of permutations of two numbers from {1, 2, 3, 4} is 12.

Using Combination: The total number of possible combinations of two numbers from this set is given by:

4C2 = 4! / (2! * (4 - 2)!) = 4! / (2! * 2!) = 6

Here, the order doesn’t matter, so we need to consider all possible combinations of two numbers from the set. Therefore, the total number of combinations of two numbers from {1, 2, 3, 4} is 6.

Conclusion

In conclusion, permutation and combination are essential concepts in statistics and mathematics, which are used to determine the total number of ways to arrange or select a specific set of objects. Permutation deals with the number of ways to select objects in a particular order, while combination deals with the number of ways to select objects without considering their order. Understanding permutation and combination is crucial in solving real-life problems and is essential for students who are attending campus interviews or preparing for competitive exams.

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