Section 3.4 Additional Topics in Probability and Counting

Factorials

Factorials are fundamental in counting problems, especially when determining the number of ways to arrange or select objects. The factorial of a non-negative integer n, denoted n!, is the product of all positive integers less than or equal to n.

• Definition: For any integer n ≥ 0, n! is defined as the product of all positive integers up to n.

• Special Cases: 0! = 1 and 1! = 1 by definition.

• Example: 5! = 5 × 4 × 3 × 2 × 1 = 120

Calculator Tip: Use the PRB (probability) button to access the factorial function (!).

Permutations (nPr)

Permutations count the number of ways to arrange r objects selected from n distinct objects, where order matters.

• Definition: An ordered arrangement of objects.

• Formula:

• Example: The number of ways to award 1st, 2nd, and 3rd place among 20 contestants: ways.

Combinations (nCr)

Combinations count the number of ways to select r objects from n distinct objects, where order does not matter.

• Definition: An unordered selection of objects.

• Formula:

• Example: The number of ways to choose 6 numbers from 52 in a lottery: ways.

Distinguishable Permutations

When some objects are identical, the number of distinguishable arrangements is less than the total number of permutations. This is important for problems involving repeated items.

• Definition: The number of ways to arrange n objects, where there are groups of identical items.

• Formula:

• Where are the counts of each type of repeated object, and .

• Example: Arranging the letters in "AABBB": ways.

Summary Table: When to Use Each Counting Method

Counting and Probability

Counting methods are often used to determine the number of possible outcomes in a sample space, which is essential for calculating probabilities.

• Sample Space: The set of all possible outcomes.

• Probability of an Event E:

• Example: Probability of guessing a unique 4-digit code (no repeats):

Worked Examples

• Committees: Number of ways to choose 4 students from 15 (order not important):

• Race Results: Number of ways 20 people can finish 1st, 2nd, 3rd (order matters):

• Arranging Disks with Repeats: 4 red, 6 black, 2 green, 3 yellow disks in a row:

• All Arrangements: 8 skiers, all finish orders:

• Lottery: Ways to choose 6 numbers from 52 (order not important):

• Probability of Winning Lottery:

• Singing Competition: 16 finalists, top 5 prizes (order matters):

• Necklace with Repeated Glass: 9 blue, 3 red, 7 green pieces:

• Pizza Toppings Probability: Probability of a specific 3-topping pizza from 9 toppings: possible pizzas,

• Cyclists Race: 15 cyclists, top 3 finishers:

• Basketball Defense Assignments: 5 players, each defends one: or

• Restaurant Meal Combinations: 3 appetizers (10 choices), 2 main courses (18), 2 desserts (13):

• Soil Sample Selection: Choose 2 contaminated from 8, 8 non-contaminated from 42:

Key Takeaways

• Use factorials for arranging all items or when r = n.

• Use permutations when order matters and selecting r from n.

• Use combinations when order does not matter and selecting r from n.

• Use distinguishable permutations when arranging items with repeats.

• Counting methods are essential for calculating probabilities in finite sample spaces.

Additional info: The notes also emphasize calculator usage for these operations, especially for large numbers, and the importance of entering expressions in one step to avoid overflow or errors.