Permutations and Combinations for JEE Mains

Aug 7, 2024

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JEE Mains 2.0: Permutations and Combinations Lecture Notes

Overview

  • Objective: Cover theory and questions on Permutations and Combinations for JEE Mains 2.0.
  • Structure: Daily 100-minute sessions, covering the latest syllabus and previous years' questions (P.Y.Qs).
  • Importance: Crucial for JEE Mains as well as for the Probability chapter.

Key Topics Covered

Basic Definitions

  • Permutations: Arrangement of elements where order matters.
  • Combinations: Selection of elements where order doesnтАЩt matter.

Fundamental Principles

  • Counting Principles: Multiplication and Addition principles.
    • If event A can happen in 'm' ways and event B in 'n' ways, together they occur in 'm ├Ч n' ways.
    • If event A can happen in 'm' ways or event B in 'n' ways, their total occurrence is 'm + n'.

Common Problems and Topics

  • Rank Problems: Using tricks to solve quickly.
  • Division into Groups: Forming groups with or without repetition.
  • Number of Combinations and Permutations: Utilizing basic principles and formulas.

Specific Problem Types

Arrangement Under Constraints

  • Example: No two boys/girls sit together.
  • General Approach: Fix positions for one group and place the other group in the remaining slots.

Fundamental Principle of Counting

  • Example: Class with 50 boys and 30 girls, forming teams with certain conditions.
  • General Approach: Use multiplication and addition principles to count possible arrangements.

Permutations with Repetition

  • Example: Forming numbers/words with given digits/letters where repetition is allowed.
  • General Formula:
    • For n different things taken 'r' at a time with repetition allowed: n^r.

Circular Permutations

  • Basic Formula: (n-1)! for arranging 'n' items in a circle.
  • With Identical Arrangements: Divide by 2 if clockwise and anti-clockwise arrangements are considered identical.

Problems on Inclusion and Exclusion Principle

  • Example: Counting elements satisfying at least one of several conditions.
  • General Formula: Use inclusion-exclusion principle to avoid over-counting.

Rank Finding of Words

  • **Steps: **
    1. Arrange letters in alphabetical order.
    2. Calculate positions by fixing letters one by one and counting permutations of the remaining letters.
    3. Adjust for repetitions.

Largest Exponent in Factorials

  • Method: Find the highest power of a prime 'p' dividing 'n!'.
  • Example: Power of 3 in 66! involves summing quotients of 66 divided by 3, 9, 27, etc.

Miscellaneous Problems

  • Number of Divisors: Use prime factorization to determine total divisors of a number.

Practice Questions

  • Provided for self-practice with links to solutions.
  • Includes a variety of question types to solidify understanding.

Resources and Links

  • P.Y.Qs: Links to last 5 years' solved questions.
  • Detailed Solutions: Available in the description box.
  • Additional Notes and PDFs: Provided upon achieving a certain number of likes.

Conclusion

  • Next Steps: Focus on Probability chapter next.
  • Feedback: Encouraged to provide feedback in the comments.

Note: 2.0 sessions are a comprehensive resource for JEE Mains preparation, covering theory, P.Y.Qs, and additional important questions.