Permutations and Combinations

Introduction

The procedures of permutation and combination are used to count the number of alternative outcomes in diverse scenarios. Permutations are arrangements, while combinations are referred to as choices. According to the fundamental principle of counting, there are sum rules and product rules to apply counting conveniently. The concept of factorials must be remembered to comprehend permutation and combination. n is equal to the product of the first n natural numbers! The number of possible arrangements for n dissimilar things is n! Let’s explore the permutation and combination study material in detail.

Permutations

A permutation is an arrangement of a number of objects in a specific order, taken one at a time or all at once. Consider the following ten numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. P(10,4) = 5040 is the number of different 4-digit-PINs that may be constructed using these 10 digits. This is a simple permutation example. Permutations of four numbers drawn from a set of ten numbers equal to the factorial of ten divided by the factorial of the difference between ten and four.

Combinations

Combinations can be used to calculate the number of possible groups that can be constructed from the available items. Examples of combinations can be divided into two categories:

Cases in which repetition is permitted, such as in the case of coins in your pocket (2,5,5,10,10)

Cases in which it is not permitted: Lottery numbers, for example, are not allowed to be repeated (2,14,18,25,30,38)

Permutation and Combination Formulas

Permutation and combination notions involve a lot of formulas. The following are two important formulas:

Formula for Permutation

A permutation is the selection of r items from a collection of n items without replacement, with the order of the items being important.

nPr = (n!) / (n-r)!

Combination Formula

A combination is a selection of r items from a set of n items with no replacements and no regard for order.

Derivation of Combination Formula:

Let us assume that there are r boxes, and each of them can hold one thing.

– Number of possible ways to choose the first thing from a set of n objects: n

– Number of methods to choose the second object from among (n-1) different options: (n-1)

– Number of methods to choose the third object from (n-2) different options: (n-2)

– Number of possible ways to choose the rth object from a set of [n-(r-1)] different objects: [n-(r-1)]

Difference Between Permutation and Combination

The permutation and combination study material will help you understand the differences between them.

• When an order/sequence of arrangement is required, permutations are utilised. When simply the number of feasible groups needs to be identified, and the order/sequence of arrangements isn’t important, combinations are employed.

• Permutations are utilised for a variety of things. Combinations are used to describe similar items.
• Ab, ba, bc, cb, ac, ca is the permutation of two items from three given things a, b, c. Ab, bc, ca is a combination of two things from three given things a, b, and c.
• It could be said that for a multitude of arrangements of things nPr=n!/(n-r)!. On the other hand, for different possible selections of things, nCr =n!/r!(n-r)!
•  The permutation response is larger than the combination answer for a given set of n and r values.

Examples on Permutation and Combination

Question 1: If n = 15 and r = 3, calculate the number of permutations and combinations.

Answer: n is equal to 15, r is equal to 3 (Given)

Using the formulas for permutation and combination, we get:

Permutation, P = n!/(n – r)!

= 15!/(15 – 3)!

= 15!/12!

= (15 x 14 x 13 x 12!)/12!

= 15 x 14 x 13

= 2730

Additionally, Combination, C = n!/(n – r)!r!

= 15!/(15 – 3)!3!

= 15!/12!3!

= (15 x 14 x 13 x 12! )/12!3!

= 15 x 14 x 13/6

= 2730/6

= 455

Question 2: An event organiser has ten chair designs and eight table patterns. How many different ways can he create a set of tables and chairs?

Answer:

The event planner features ten chair patterns and eight table patterns.

There are ten different ways to choose a chair.

A table can be chosen in eight different ways.

As a result, one chair and one table can be chosen in ten different ways.

= 80 different ways

Conclusion

After going through the study material on Permutations and Combinations, it can be seen that they both are used in a variety of contexts. Permutations are counting used to arrange r different objects out of n different objects. Combinations are a type of counting used to select r different objects from a set of n different objects. The permutation is the number of combinations created by selecting r number of items from a set of n items. The number of possible groups of r objects built from the available n objects is the combination. You can refer to permutations and combinations of study material for more information.