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Permutation and Combination – Basics – Introduction

Fundamental of counting, permutations and combinations are covered in this blog. Start you journey into the world of combinatorics by learning the basics. Few theorems included in the blog.

5 minutes long
Posted by Tanwir Silar, 11/8/2021
Permutation and Combination – Basics – Introduction

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In this blog, learn about the fundamental principle of counting and how to count the number of permutation and combination in a easier manner.

Fundamental Principle of Counting

There are two rules of counting. The multiplication rule and the addition rule.

Multiplication rule

For a task to be completed, two events A, B need to be performed. If the event A can occur in m different ways, following which event B can occur in n different ways, then the total number of ways in which task can be completed is m x n.
For instance, the number of ways a person can dress who has 3 pants and 2 shirts is 3 x 2 = 6.

Addition rule

For a task to be completed, either of the two events A, B should be performed, but not both. If the event A can occur in m different ways, and event B can occur in n different ways, then the total number of ways to complete the task is m+n.

What is a Permutation?

A permutation is an arrangement in a definite order of a number of objects taken some or all at a time. The number of permutations of n different objects taken r at a time, when repetition is not allowed is Permutation notation short, when repetition is allowed is nk.

What is a Combination?

A combination is an arrangement in which the order of elements arranged doesn’t matter. The number of combinations of n different objects taken r at a time, when repetition is allowed is Combination notation short, when repetition is not allowed is Choosing k from n+k-1 items.

Factorial notations

Factorials come handy to calculate permutation and combination. Mainly because,
Combination notation
Permutation notation.
Getting familiar with factorial notation comes in handy while solving problems of Permutation and Combination.

Permutation and Combination Theorem 1

Theorem 1:
Let there are n objects. In these n objects we have p1 objects of one kind, p2 objects of second kind, … pk objects of kth kind and the rest if any, are of different kind. Then the number of permutations of the n objects taken all at a time equals to image for theorem 1

Permutation and Combination Theorem 2

Theorem 2:
This theorem tells us how permutation and combination of n different objects taken r at a time, when repetition is not allowed are related as Relation between npk and nck
This theorem tells us that to find the number of permutations we can find the number of combinations and multiply it with r! ( use the result carefully, as certain conditions apply).

Example:
Find the number of combinations and permutations of choosing 4 cards out of a 52 card deck?
As discussed earlier, the number of combinations are 52 choose 4 = 52 choose 4
Using the above theorem, we could easily get the number of permutations from number of combinations.
In continuation, the number of permutations are =


permutations of 52 objects taken 4 at a time

We hope, this blog(permutation and combination) was helpful.



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