Combinations

Definition: Combinations are a fundamental concept in combinatorics that deal with the selection of objects or elements from a set without regard to the order of selection. In combinations, the order of the selected elements does not matter.

Key Concepts:

  1. Combination: A combination is an unordered selection of "r" distinct objects from a set of "n" distinct objects. In combinations, the order of selection is not considered.

  2. Binomial Coefficient Notation: Combinations are often represented using binomial coefficient notation. The binomial coefficient "C(n, r)" represents the number of combinations of "n" items taken "r" at a time and is calculated using the formula:

    C(n, r) = n! / [r! * (n - r)!]

    Where:

    • C(n, r) represents the number of combinations.
    • n! denotes the factorial of "n."
    • r! denotes the factorial of "r."
    • (n - r)! denotes the factorial of "n - r."
  3. Example: Consider a set of 5 distinct books, and you want to select 3 books to read. To find the number of combinations, use the combination formula:

    C(5, 3) = 5! / [3! * (5 - 3)!] C(5, 3) = (5 × 4 × 3 × 2 × 1) / [(3 × 2 × 1) * (2 × 1)] C(5, 3) = 10 combinations

    So, there are 10 different ways to select 3 books out of 5, regardless of the order of selection.

  4. "n choose r": Combinations are often referred to as "n choose r," indicating that you are choosing "r" items from a set of "n" items. For example, "5 choose 3" means selecting 3 items from a set of 5.

  5. Properties of Combinations:

    • Combinations are denoted by "C(n, r)" or "n choose r."
    • Combinations are equal for "n choose r" and "n choose (n-r)," meaning that choosing "r" items from a set is the same as choosing "n-r" items, as the order does not matter.
    • The number of combinations is always a non-negative integer.
  6. Applications: Combinations are widely used in various fields, including statistics, probability theory, and decision-making. They help determine the number of ways to select groups of items or individuals from a larger set, where the order of selection is not relevant.

  7. Combinations with Repetition: In some cases, combinations with repetition are used when items can be selected multiple times. The formula for combinations with repetition is:

    C(n + r - 1, r) = (n + r - 1)! / [r! * (n - 1)!]

    Where:

    • n represents the number of distinct items.
    • r represents the number of selections.
    • The formula accounts for repetitions and the order of selection remains unimportant.

Key Concepts

  • Order Does Not Matter: In combinations, the order of the selection does not matter. For example, the combinations of the set {1, 2, 3} taken 2 at a time are {1, 2}, {1, 3}, and {2, 3}.
  • Repetition: Combinations can be with or without repetition. In combinations with repetition, an object can be used more than once in the selection. In combinations without repetition, each object can only be used once in the selection.

Examples

  • Example 1: How many ways can 3 people be chosen from a group of 10 people? The number of combinations of 10 people taken 3 at a time is C(10, 3) = 10! / (3!(10 - 3)!) = 120.
  • Example 2: How many ways can 2 letters be chosen from the word "APPLE"? The number of combinations of 2 letters taken from the word "APPLE" is C(5, 2) = 5! / (2!(5 - 2)!) = 10.
  • Example 3: In a group of 8 people, how many ways can a committee of 5 people be chosen? The number of combinations of 5 people taken from a group of 8 people is C(8, 5) = 8! / (5!(8 - 5)!) = 56.