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.