9. Combinations and Permutations

Combinations: Choosing multiple items from multiple groups using the word "or" - example:

"Or" implies that the events are mutually exclusive (e.g., they cannot occur at the same time) We have to add the number of possible ways that event A can occur to the number of ways that event B can occur when events A and B are mutually exclusive For mutually exclusive events (i.e., they can be selected independently, we have to add the number of unique possibilities by determining the total number of possibilities for each of the options)

Solving combination problems

1. Basic Combination Formula (aka "n choose k") - Screenshot - Variable "n" represents the number of objects from which we will choose - Variable "k" represents the number of objects we will actually choose - Example: If we are to select 5 items out of a pool of 10 items, then we refer to the problem as 10 choose 5 N Choose K = N! / [(N-K)! x K!] 2. Box and Fill Method

Permutations with Restrictions - Anchor Method:

Anchor Method: - Anchor the items in place (e.g., 5 dogs must be lined up but you know that Zhao the dog will be 1st in line. Therefore there are 4! ways to line up the rest of the 4 dogs) Link Method: - Link together items that should be considered as 1 unit, and multiply the end result by (X) with X representing the number of items that are being linked together (since they can be ordered separately within their linked box together) - Example (SS) in the next flashcard below

Introduction

Both combinations and permutations refer to methods of counting the number of ways in which a certain task can be completed. Combinations are used when the order in which a task is completed does not matter. Permutations, in contrast, are used when the order in which a task is completed does matter.

Combinations: Choosing multiple items from multiple groups using the word "and" - example:

For "and" problems where we have to choose option #1 from a group and option #2 from another group - We multiply the total number of possibilities from each group together. As such, solution = Total number of possibilities from option #1 (out of Group A) x Total number of possibilities from option #2 (out of Group B)

Combinations and Permutations: Some items can never be together in the same subgroup

If two events, A and B occur in a scenario and are both collectively exhaustive and mutually exclusive, it must be true that The total number of ways in which Scenario can occur = The number of ways in which A can occur + The number of ways in which B can occur Example question: - Three people are to be selected from a pool of 5 people to form a writing club. If two of the people can never be together in the club, in how many different ways can the club be formed? Solution (screenshot): - First solve for total number of ways to form the club if there are no restrictions - Then we solve for the # of possibilities in which two people are together (in order to subtract it from the total number of possibilities to determine the # of possibilities in which they CANNOT be together); this is a 3 choose 1 type situation since there is only 1 spot remaining if we are going to assume two people are together already (meaning there are only 3 people left in the pool and only 1 "decision" left to make) - Therefore, 10 - 3 = 7

Circular Permutation Formula

Number of Ways to Arrange a Set of Items in a Circle = (N - 1)! N represents the number of items to be arranged in a circle (e.g., 6 items arranged in a circle is 5! number of ways to arrange a set of items)

Permutations - Two Methods to Solve:

Permutation Formulas: 1. Basic Permutation Formula (aka "n pick k") N Pick K = N! / (N - K)! 2. Box and Fill Method - Same as for combinations, except now we do not divide by N factorial at the bottom of the equation

Basic Combination Formula Example Question

Question: - A coach must select 3 softball players from a pool of 7 possible players to play for his team. How many different ways are available to the coach in selecting these 3 players?

Combinations: Box and Fill Method - Example (Same as above)

Question: - A coach must select 3 softball players from a pool of 7 possible players to play for his team. How many different ways are available to the coach in selecting these 3 players? Answer: - Have 1 box for each decision that has to be made and then the number of options for each decision should be placed in that box - Divide that by the factorial of the number of decisions

Combinations

Refers to methods of counting the number of ways in which it is possible for a task to be accomplished when the particular order in which the task is accomplished does not matter - e.g., select 5 candies from a dish containing 10 different colored candies

Permutations - Some Items Can't Be Next to Each Other:

Screenshot - Treat it like a mutually exclusive problem Solve for total number of ways to arrange minus the total number of ways to arrange with them standing next to each other

Creating Codes

Screenshot (self-explanatory)

Combinations: Choosing "at least" some number of items

The phrase "at least" means equal to or greater than. "At least" problems typically involve the addition of outcomes since they will be considered as mutually exclusive events (e.g., purchase at least 6 items, means we can purchase 1, 2, 3, 4, etc.) Example - screenshot: - We have to add all of the possible different outcomes together - We have to manually determine how many possible outcomes there are (which should be simple). For example, if we have to choose at least 4 people for a team of 6 from China, then there can be 4, 5, or 6 people from China on that team. - Following from that, we have to add the # of ways that 4 people from China can be selected from the broader group + # of ways that 5 people can be selected + # ways that 6 people can be selected.

Combinations: Fundamental counting principle

Uses multiplication of the number of ways each event in an experiment can occur to find the number of possible outcomes in a sample space. e.g., 5 types of eggs, 3 types of coffee, and 6 types of pastries means there are 5 x 3 x 6 = 90 different possible combinations

Combinations with Restrictions: When Some Items Must Be Chosen (or must not be chosen)

We should essentially consider the items which must be chosen as having been eliminated from the pool of items Example: If there is a pool of 20 people from which we have to select a 6-person committee, and Jeff and Cali must be selected, then there is actually only a pool of 18 people and we have to select a 4-person committee - Then we just solve based on the pool of 18 people and 4-person committee If there are items which must NOT be chosen, then we simply remove those people from the main pool of people when performing our calculation Main trick here is to immediately remove the objects which should be included, or include the people who have to be included as part of the equation. - Do this before starting any calculations or applying any formulas.

Permutation Formula for Indistinguishable Items

e.g., in a set of "SSSSS", each S is indistinguishable from the other, so there is really only 1 way to arrange the set Example (SS)