Lecture on Combinatorial Problems and Number Theory in IIT-JEE

Key Topics Covered:

• LCM of prime numbers in combinatorial problems
• Identical object selection
• Divisors in number theory
• Counting principle in matrix problems
• Non-coplanar vectors in combinatorial geometry

Combinatorial Problem Involving Prime Numbers

• Given RST as prime numbers; P and Q as positive integers
• LCM of P and Q is R^2 * T^4 * S^2
• Distribution of factors (R, T, S) important for LCM calculation

Methods to solve:

1. Distribution of R factors:
   • Cases: (R^0, R^2), (R^1, R^1), (R^2, R^0)
   • Total possibilities: 2 * 3 - 1 = 5
2. Distribution of T factors:
   • Cases: T^0 to T^4
   • Total possibilities: 2 * 5 - 1 = 9
3. Distribution of S factors:
   • Total possibilities: 2 * 3 - 1 = 5

Total ordered pairs P and Q: 5 * 9 * 5 = 225

Identical Object Selection

• n identical objects: ways to select one - always 1
• Selecting k objects from n identical objects:

Key Points:

• At least one object: ways = n
• 0 or more objects: ways = n + 1

Example Problem: Selection of Identical Balls

1. White balls (10), Green balls (9), Black balls (7)
2. Selection principle:
   • Count each color's possibilities separately
   • Total number of examples using combinatorics
   • Subtract 1 to exclude no selection case
   • Final total: (11) * (10) * (8) - 1 = 879

Number of Divisors

• Example: 24 = 2^3 * 3
• Number of divisors calculation:
  • Select subsets from prime factors
  • For each prime factor, include 0 to max power possibilities
  • General formula: (p+1)(q+1)(r+1)...

Special cases:

• Even divisors: at least one 2-powers
• Odd divisors: exclude all 2-powers

Matrix Problems in Counting Principle

• Example from JEE 2010:
  • Symmetric/Skew-symmetric matrices
  • P number of symmetric matrices
  • Cases: A not 0 and determinant divisible by P

Special matrices from set properties

1. Perfect square divisors
   • Only even occurrence of prime powers
2. Eigenvalues or given condition-specific problems

Advanced Problem Application

1. JEE Advanced 2013:
   • Set V of vectors with elements in {-1, 1}
   • Non-coplanar vector selection
   • Select triplets ensuring non-coplanarity using combination logic

Takeaways

1. Combinatorial principles crucial in number theory problems
2. Logical breakdown of complex problems aids in simplifying solutions
3. Repeated practice with JEE-comparable models enhances problem-solving skills