Overview
This lecture covers the fundamental rules and examples of permutations and combinations, focusing on counting principles, factorials, and problem-solving strategies for arranging and selecting items.
Basic Counting Principles
- The multiplication principle (m × n): Multiply the number of options for each choice to get the total number of outcomes.
- The power principle (a^b): When filling b spaces, each with a choices, use a to the power of b.
- The factorial principle (n!): Used for arranging n objects in order with no restrictions.
Factorials
- n factorial (n!): n × (n-1) × (n-2) ... × 1.
- 0! = 1 by definition.
- Example: 5! = 5 × 4 × 3 × 2 × 1 = 120.
Permutations
- Denoted as nPr (n is total, r is selected/arranged).
- Used when order matters; formulas: nPr = n!/(n−r)!.
- Example: 8P5 = 8!/(8−5)! = 6720.
Combinations
- Denoted as nCr (n is total, r is selected).
- Used when order does not matter; formula: nCr = n!/(n−r)!r!.
- Example: 10C3 = 120.
Key Examples
- Arranging 5 boys and 4 girls where first two must be girls: 5 × 4 × 7! = 100,800.
- Arranging "HISTORY": 7! = 5040 ways.
- Arrangements where specific letters are together: Treat the group as one block and multiply by their arrangement.
- Calculating cases where arrangement/selection is restricted: Subtract restricted cases from total.
Problem Solving Strategies
- For selection problems (“appoint” or “choose”), use combinations.
- For arrangement problems (“arrange” or “order”), use permutations.
- When specific items must or must not be together, use grouping or subtraction methods.
Key Terms & Definitions
- Permutation — Arrangement of objects where order matters.
- Combination — Selection of objects where order does not matter.
- Factorial (n!) — Product of all positive integers up to n.
- nPr — Number of permutations (order matters) of n items taken r at a time.
- nCr — Number of combinations (order doesn’t matter) of n items taken r at a time.
Action Items / Next Steps
- Practice additional permutation and combination problems.
- Review how to use your calculator’s nPr and nCr functions.
- Prepare questions for the next lecture on more complex permutation and combination problems.