## Dr. J’s Guide to Combinations (without replacement)

Dr. J introduces combinations (without replacement) which is an unordered set of k elements taken from a set of n elements where elements cannot be repeated. He provides intuition behind the formula for combinations, which is n factorial divided by (n-k) factorial times k factorial. Namely, combinations (without replacement) are just permutations (without replacement) except …

## Dr. J’s Guide to Permutations (without Replacement)

Permutations (without replacement) are an ordered set of k elements taken from a set of n elements where elements cannot be repeated. In constructing this set, we have n choices for the first element, (n-1) choices for the second element, and so on until we have (n-k+1) choices for the kth element. Thus the formula …

## Dr. J’s Guide to Permutations with Replacement

Dr. J introduces delves into more detail on permutations with replacement including intuition behind the formula and additional examples. To aid the intuition, the Fundamental Rule of Counting is introduced that states “if there are a ways of doing one thing and b ways of doing another thing, then there are a times b ways …

## Dr. J’s Guide to Counting

Dr. J introduces the four basic types of counting: combinations and permutations both with and without replacement. Combinations occur when order does not matter while permutations occur when order does matter. The video shows examples of using formulas to perform the counting for each the four basic types of counting.

## Dr. J’s Guide to Probability

Dr. J introduces the mathematical definition of probability. An experiment is any data collection process. A particular result of an experiment is an outcome. The collection of all possible outcomes is the sample space of the experiment. Any subset of the sample space is an event. Probability is a function of an event. By Kolmogorov’s …

## Dr. J’s Guide to Cardinality

Dr. J introduces the idea of cardinality, the size of a set. Three categories are discussed: finite, countably infinite, and uncountably infinite. Sets with finite size are those where you can count the elements and eventually stop. Sets with countably infinite size can be put into one-to-one correspondence with the natural numbers. Sets with uncountably …

## Dr. J’s Guide to Partitions

Dr. J introduces a property the set properties of disjoint, pairwise disjoint, and partition. Two sets are disjoint if their intersection is the empty set. A collection of sets is pairwise disjoint if every pair of sets is disjoint. A collection of sets is a partition of S if 1) none of the sets is …

## Dr. J’s Guide to Venn Diagrams

Dr. J introduces Venn Diagrams which are a visual illustration of sets. Venn Diagrams are then used to visually demonstrate subsets and supersets as well as the set operations of union, intersection, set difference, and complement.

## Dr. J’s Guide to Set Operations

Dr. J introduces set operations including union, intersection, set difference and set complement. The union of two sets contains all the elements in either set. The intersection of two sets contains the elements that are in both sets. The set difference of A minus B contains all the elements in A that are not in …

## Dr. J’s Guide to Subsets & Set Comparisons

Dr. J introduces the idea of a subset and set comparisons. Set A is a subset of set B if the elements of A are all in the set B. The set A is a proper subset of B if there is at least one element in B that is not in A. Thus subset …