### 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.

**Reference:**

- NCERT Maths Textbook for Class 11 and Class 12