Lecture Notes: Permutations and Combinations

Introduction

• Permutations and Combinations: Fundamental concepts used to solve problems involving arrangement and selection.
• Lecture Objective: Cover key aspects of permutations and combinations, solve problems, and understand when to apply each concept.

Permutations

• Definition: Arrangement of objects where order matters.
• Examples:
  • Arranging books, people in line, letters in a word, and numbers.
  • Key Idea: If changing the order results in a different outcome, it is a permutation problem.
• Calculation:
  • Use factorial notation (e.g., n!) to determine the number of ways to arrange n items.
  • Use nPr for permutations of r items from n available.

Permutations with Restrictions

• Permutations and Combinations:
  • Solve problems like arranging people/objects with specific conditions (e.g., certain items must be together).
• Concept of Blocks:
  • Group items that must remain together into a single block.
  • Calculate permutations of the block and remaining items.

Examples of Permutations

• Basic Arrangements:
  • Simple cases like arranging three people in a line.
  • Use of factorial to calculate (e.g., 3! = 6).
• More Complex Scenarios:
  • Items with repetition (e.g., "BOB" where B is repeated).
  • Calculate using division by factorial of repeated items.
  • Example: "trigonometry" involves multiple repeated letters; adjust calculation accordingly.

Special Cases

• Permutations with Fixed Positions:
  • When certain positions are fixed or restricted.
  • Handle conflicts by separating cases (e.g., specific digits in a number).
• Handling Conflicts:
  • Separate cases for conflicting items (e.g., same digit cannot be in two places).
  • Example: Six-digit number formation with conflict between first and last digits.

Combinations

• Definition: Selection of items where order does not matter.
• Examples:
  • Selecting a committee, team, or subset of items.
  • Key Idea: AB is the same as BA, only the identity of the selected items matters.
• Calculation:
  • Use nCr to calculate combinations of r items from n available.

Combinations in Different Contexts

• Selecting Teams or Committees:
  • Choose items from distinct categories (e.g., Runners, Sprinters, Jumpers).
  • Multiply selections from each category for total combinations.
• At Least or No More Than Conditions:
  • Use logic to determine combinations that meet specific criteria (e.g., at least one of an item, more men than women).

Examples of Combinations

• Basic Selection Problems:
  • Selecting photographs, books, or people without additional conditions.
  • Calculate using direct application of nCr.
• Handling Restrictions and Conditions:
  • Solve problems with specific restrictions (e.g., T must be selected, more men than women).
  • Consider multiple scenarios to meet the same condition (e.g., various ways to have more men than women).

Conclusion

• Summary:
  • Understanding when to use permutations vs. combinations is crucial.
  • Practice solving problems with different restrictions and conditions.
• Next Steps:
  • Continue practicing with worksheets and additional problems.
  • Attend next session for further examples and detailed explanations.