SATHEE: Notes from Toppers

Permutation and Combination - Detailed Notes

1. Fundamental Principles:

Permutation: An arrangement of objects in a definite order.
- Formula: $$nPr = n! / (n - r)!$$
Combination: A selection of objects without regard to order.
- Formula: $$nCr = n! / (r!(n - r)!) $$
Factorial: The product of all positive integers up to a given number.
- Formula: $$n! = n \times (n-1) \times (n-2) \times \dots \times 2 \times 1$$

2. Counting Techniques:

Permutation Formula (nPr): If there are n objects arranged in a specific order, then there are nPr possible permutations of those objects.
Combination Formula (nCr): If there are n objects and we want to select r objects, then there are nCr possible combinations of those objects.
Permutation with Repetition: If there are n objects and each object can be repeated any number of times, then there are n^r possible permutations of those objects.
Combination with Repetition: If there are n objects and each object can be repeated any number of times, then there are (n+r-1)Cr possible combinations of those objects.
Circular Permutation and Combination: If there are n objects arranged in a circle, then there are (n-1)! possible circular permutations of those objects. Similarly, there are (n-1)Cr possible circular combinations of those objects.

3. Applications in Probability

Permutations and combinations can be used to calculate the probability of an event occurring.
The probability of an event is the ratio of the number of favorable outcomes to the total number of possible outcomes.

4. Derangements

A derangement is a permutation in which no element remains in its original position.
The number of derangements of n objects can be calculated using the formula: $$ D_n = n! - n \cdot (n-1)! $$

5. Inclusion-Exclusion Principle

The inclusion-exclusion principle is a counting technique that allows for the calculation of the number of elements in a set by including and then excluding certain subsets.
The principle states that; if A and B are two finite sets, then $$|A \cup B| = |A| + |B| - |A \cap B|$$

6. Ordered and Unordered Arrangements

An ordered arrangement is an arrangement in which the order of the elements matters.
An unordered arrangement is an arrangement in which the order of the elements does not matter.
For n elements, there are n factorial (n!) ordered arrangements and n-1 factorial (n-1)! unordered arrangements.

7. Applications in Geometry and Algebra

Permutations and combinations can be used to solve various problems geometry and algebra.
For example, they can be used to determine the number of different ways in which a set of points can be arranged in a plane or the number of solutions of a system of linear equations.

8. Mathematical Reasoning and Problem Solving

Permutations and combinations require strong mathematical reasoning and problem-solving skills.
To excel in this topic, it is important to develop the ability to analyze and solve complex counting problems.

9. Formulae and Identities

There are a number of important formulae and identities related to permutations and combinations.
It is important to memorize these formulae and be able to apply them efficiently.

10. Advanced Concepts

Advanced concepts in permutations and combinations include generating functions, recurrence relations, and the pigeonhole principle.
These concepts can be used to solve more challenging counting problems.

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