Permutations and Combinations

Permutations

Permutations are used to count the number of ways to arrange objects where order matters. The concept is fundamental in probability and combinatorics, and is widely applied in statistics for counting possible outcomes.

• Definition: A permutation is an ordered arrangement of objects. The number of different permutations of n distinct objects is n! (n factorial).

• Factorial: For a positive integer n, n! is defined as the product of all positive integers up to n: By definition, .

• Example: The number of ways to arrange the first row of a Sudoku grid (9 digits):

• Permutations of n Objects Taken r at a Time: The number of ways to arrange r objects selected from n distinct objects is:

• Example: The number of four-digit codes with no repeated digits (selecting 4 from 10):

• Example: Number of ways 33 race cars can finish first, second, and third:

• Distinguishable Permutations: When some objects are identical, the number of distinguishable permutations is: where are the counts of each type.

• Example: Arranging the letters in AAAABBC (4 A's, 2 B's, 1 C):

• Example: Arranging 6 one-story, 4 two-story, and 2 split-level houses:

Combinations

Combinations are used to count the number of ways to select objects where order does not matter. This is essential in probability calculations where the arrangement is irrelevant.

• Definition: A combination is a selection of objects without regard to order. The number of combinations of n objects taken r at a time is:

• Example: Selecting 3 beaches out of 5 for restroom construction:

• Example: Selecting 4 companies from 16 bidders:

• Example: Forming a 3-person committee from 20 employees:

Applications of Counting Principles

Summary Table of Counting Principles

Choosing the Appropriate Counting Principle

• Are there two or more separate events? Use the Fundamental Counting Principle.

• Is the order of the objects important? Use Permutations.

• Is order not important? Use Combinations.

• Some problems may require more than one principle.

Applications to Probability

Using Counting Principles to Find Probabilities

Counting principles are often used to determine the number of possible outcomes (the sample space) and the number of favorable outcomes, which are then used to calculate probabilities.

• Probability Formula:

• Example: Probability of selecting 3 specific members for chair, secretary, and webmaster from 17:

  • Number of ways to assign positions:

  • Probability:

• Example: Probability of being dealt 5 diamonds from a standard deck:

  • Ways to choose 5 diamonds:

  • Total 5-card hands:

  • Probability:

• Example: Probability of selecting exactly one toxic kernel from 4 chosen out of 400 (3 toxic):

  • Ways to choose 1 toxic:

  • Ways to choose 3 nontoxic:

  • Total ways:

  • Total possible:

  • Probability:

Real-World Applications

• Lottery Probability: Calculating the probability of winning the Powerball lottery involves combinations:

  • Old rules: Choose 5 numbers from 59 and 1 from 35.

  • New rules: Choose 5 numbers from 69 and 1 from 26.

  • Probability is the reciprocal of the total number of possible combinations.

• Committee Selection: Choosing a committee from a group uses combinations, as order does not matter.

• Arranging Objects: Assigning positions or arranging items where order matters uses permutations.

Technology Tips

• Most scientific calculators and statistical software (e.g., TI-84 Plus, Excel, Minitab, StatCrunch) have built-in functions for permutations and combinations.

• On the TI-84 Plus, use the PRB menu to access nPr and nCr functions.

• In Excel, use PERMUT(n, r) for permutations and COMBIN(n, r) for combinations.

Summary

• Permutations and combinations are essential tools for counting and probability in statistics.

• Permutations are used when order matters; combinations are used when order does not matter.

• Distinguishable permutations account for identical objects.

• Counting principles are foundational for calculating probabilities in a variety of real-world and theoretical contexts.