Permutations and Combinations

Permutations of Distinct Items

• Permutations refer to the number of ways to arrange a set of items in a specific order.
• When dealing with distinct items, the order matters.
• The number of permutations of n distinct items taken r at a time is denoted as P(n, r) or ⁿP_r.
• The formula for calculating permutations is P(n, r) = n! / (n-r)!, where n! represents the factorial of n.
• Example: Finding the number of ways to arrange 5 distinct books on a shelf. P(5, 5) = 5! / (5-5)! = 5! / 0! = 120 / 1 = 120

Combinations of Items

• Combinations refer to the number of ways to select a set of items from a larger set, where the order in which the items are selected does not matter.
• The number of combinations of n distinct items taken r at a time is denoted as C(n, r) or ⁿC_r or _nC_r.
• The formula for calculating combinations is C(n, r) = n! / (r! * (n-r)!).
• Example: Selecting 3 books out of 5 to take on a trip. C(5, 3) = 5! / (3! * (5-3)!) = 5! / (3! * 2!) = 10

Basic Principles of Counting

• The fundamental counting principle states that if one event can occur in m ways and a second event can occur in n ways, then the two events can occur in sequence in m × n ways.
• This principle extends to multiple events.
• Example: Choosing a shirt and pants from a wardrobe. If there are 3 shirts and 4 pants, there are 3 × 4 = 12 possible outfit combinations.

Factorial Notation

• Factorial notation (denoted by !) represents the product of all positive integers less than or equal to a given positive integer.
• n! = n × (n-1) × (n-2) × ... × 2 × 1
• Example: 5! = 5 × 4 × 3 × 2 × 1 = 120
• 0! is defined as 1.

Applications in Probability

• Permutations and combinations are fundamental to calculating probabilities.
• Determining the likelihood of specific outcomes in events often involves these concepts.
• Example: Calculating the probability of winning a lottery by correctly selecting a certain number of numbers.
• Calculating probabilities of events where order isn't a factor.

This quiz covers the concepts of permutations and combinations in mathematics. You will learn how to calculate the number of ways to arrange distinct items and how to select items without regard to order. Test your understanding of the formulas and examples provided.