Topic 1: Interpreting a tree diagram
Topic 2: Introduction to the counting principle
Problem 1: You have 3 shirts and 2 pairs of pants. Use the counting principle to find the number of outfits possible.
Answer: Number of outfits = 3 shirts × 2 pants = 6.
Problem 2: There are 4 books and 3 shelves. Calculate the number of ways to place one book on one shelf using the counting principle.
Answer: Number of ways = 4 books × 3 shelves = 12.
Topic 3: Counting principle
Problem 1: A menu has 5 appetizers, 4 main courses, and 3 desserts. Find the number of possible meals (one of each).
Answer: Number of meals = 5 × 4 × 3 = 60.
Problem 2: You can choose 2 colors from 4 options and 1 size from 3 options for a shirt. Calculate the total combinations.
Answer: Total = 4 colors × 3 sizes = 12 (assuming order doesn't matter for colors, but counting principle applies to sequential choices).
Topic 4: Introduction to permutations and combinations
Problem 1: Explain the difference between permutations and combinations using arranging 3 books on a shelf (permutation) vs. choosing 3 books from 5 (combination).
Answer: Permutations consider order (e.g., ABC, ACB are different; P(3,3) = 6). Combinations do not (e.g., {A,B,C} is one; C(5,3) = 10).
Problem 2: For selecting 2 officers from 4 people, distinguish if order matters (permutation) or not (combination).
Answer: If order matters (president, vice), it's a permutation: P(4,2) = 4 × 3 = 12. If not, combination: C(4,2) = 6.
Topic 5: Permutations and combinations: Problem type 1
Problem 1: Calculate the number of permutations of 4 distinct letters: A, B, C, D.
Answer: P(4,4) = 4! = 24.
Problem 2: Find the number of ways to arrange 3 people in a line from 5 people.
Answer: P(5,3) = 5 × 4 × 3 = 60.
Topic 6: Permutations and combinations: Problem type 2
Problem 1: Calculate the number of combinations of 6 items taken 3 at a time.
Answer: C(6,3) = 6! / (3! (6-3)!) = (720) / (6 × 6) = 20.
Problem 2: Find the number of ways to choose 4 books from 7 without regard to order.
Answer: C(7,4) = 7! / (4! 3!) = (5040) / (24 × 6) = 35.