Lecture Notes: Introduction to Counting and Permutations

1. Introduction to Counting

• Fundamental Counting Principle: If an event can occur in M ways and another in N ways, they can occur together in M x N ways.
• Example: Flipping a coin (2 ways: heads or tails) and rolling a die (6 ways: 1-6) results in 2 x 6 = 12 possible outcomes.

2. Permutations

• Definition: Permutations are arrangements of items where order matters.
• Formula: nPr = n! / (n-r)!, where n is the total number of items and r is the number of items to arrange.
• Factorial (n!): Product of all positive integers up to n.
• Example: Arranging 7 distinct rides at Disneyland results in 7! = 5040 different ways.

3. Permutations with Non-Unique Items

• Concept: When items are not unique, certain arrangements do not change the overall picture.
• Formula: Total arrangements = n! / (n1! x n2! x ... nk!), where n1, n2, ... are the counts of non-unique items.
• Example with Word: The word "STATISTICS" has 10 letters, with repetitions (S:3, T:3, I:2). Different arrangements can be calculated as 10! / (3! x 3! x 2!).

4. Combinations

• Definition: Combinations are selections of items where order does not matter.
• Formula: nCr = n! / [r! (n-r)!]
• Example: Selecting 5 rides out of 25 at Magic Mountain, where the order of selection does not matter, is calculated using combinations.

5. Calculation Techniques

• Using Calculators: Use the calculator functions (nPr, nCr) to quickly compute permutations and combinations.
• Steps for Calculator:
  • Enter n (total number of items).
  • Select PRB (probability functions) and choose nPr or nCr depending on whether order matters.
  • Enter r (number of items to select/arrange).

6. Real-World Applications

• Pin Number Example: Calculating possible pin numbers using the counting rule.
• Governor Visit Example: Choosing counties to visit as a permutation problem.

Summary

• Understanding the difference between permutations (order matters) and combinations (order does not matter) is crucial in counting problems.
• The fundamental counting principle helps in understanding basic counting scenarios.