Lecture: Problem-Solving in Counting & Combinatorics

Introduction

• Hello everyone, good evening! Today's session is going to be amazing and fun.
• We will discuss counting-based questions in combinatorics.
• No special formulas will be used, only logic and brainpower are needed.
• All questions will be similar to those found in IIT-JEE.
• There will be no repeating questions.

Resources & Connectivity

• Discussed Telegram ID and playlists (23 and master playlist).

Awards & Achievements

• Notable students: Prakash, Samman (International Chemistry Olympiad) & Dhruv (International Physics Olympiad).

Importance of Counting Principles

• Basic concepts of counting: Subjob 1 and Subjob 2.
• Simple counting formula: M * N
• Example: Routes from Chandigarh to Delhi, Mumbai.
• Time to apply logic.

Problem-Solving Examples

1. Counting Three-Digit Numbers

   • Question: Counting three-digit numbers that have at least one zero and one five.
   • Discussed example & solution process.
   • Overcounting error and its correction.
   • Final answer: 388

2. Counting Integral Solutions

   • Question: Number
   • Example: 4B + 5C = 54000
   • Solution process: Limited solutions
   • Observation: Solving through prime factorization.
   • Final answer: 16

3. Card Arrangement Problem

   • Analyzed and found that no card can move to more than one position.
   • Final answer: 256

4. Digit Counting for Divisibility by 3

   • Example: Numbers 21, 31, 41, 51, 71, 81
   • Solution process: Analyzing mod 3 of the number.
   • Final answer: 164

5. Various Counting Problems Related to Specific Constraints

   • Analysis of different conditions.
   • Solution process and method used.

Conclusion

• Fun and thought-provoking questions requiring the application of logic.
• Suggested students practice and learn.
• Recap of the last session and discussion of upcoming sessions.