Fundamental Counting Principle
1 hr 17 min 15 Examples
- What is the Multiplication Rule? (Examples #1-5)
- Overview of the Sum Rule and the Principle of Inclusion-Exclusion? (Examples #6-7)
- Determine the number of subsets (Examples #8-9)
- How many three-digit or four-digit integer numbers end with zero (Example #10)
- Use multiplication rule and sum rule to find the number of arrangements (Examples #11-12)
- How many four-letter arrangements can be made if repetition is not allowed (Example #13)
- On a multiple choice test how many ways can a student answer? (Example #14)
- Discover the number of four-card sequences from a standard deck (Example #15a-e)
Permutation
1 hr 16 min 13 Examples
- Permutation formulas: with or without repetition (Examples #1-3)
- How many distinguishable permutations? (Example #4a-b)
- Circular Rule for Permutations (Examples #5-6)
- Determine the number of function if f is injective (Example #7a-b)
- How many ways to arrange people around two circular tables (Example #8)
- Arrange knights at a round table if two nights refuse to sit next to each other (Example #9)
- How many tennis matches can be arranged? (Example #10)
- Use permutations to find the number of arrangements (Examples #11-13)
Combinations
1 hr 18 min 12 Examples
- Combinations without repetition (Examples #1-4)
- Combinations with repetition (Examples #5-6)
- Determine the number of bridge hands dealt from a standard deck (Example #7a-b)
- What is the minimum value of n for the subset (Example #8a-b)
- Combinations using “at most” and “at least” (Examples #9-10)
- How many poker hands can be made with at least one card from each suit (Example #12-a-b)
- How many poker hands contain exactly one pair or three of a kind (Example #12c-d)
- How many poker hands contain a full house (Example #12e)
Pigeonhole Principle
48 min 11 Examples
- What is the pigeonhole principle? (Examples #1-4)
- Generalized formula for the pigeonhole principle (Examples #5-8)
- How many cards must be selected to guarantee at least three hearts (Example #9a-b)
- Prove there are at least 7 dice with the same number in the game of TENZI (Example #10)
- Show there are at least two with the same remainder (Example #11)
Binomial Coefficient
1 hr 34 min 10 Examples
- How to use the Binomial Theorem (Example #1)
- Pascal’s Triangle for finding binomial coefficients (Examples #2-3)
- Find the indicated coefficient for the binomial expansion (Examples #4-5)
- Find the constant term of the expansion (Examples #6-7)
- Binomial theorem to find coefficients for the product of a trinomial and binomial (Examples #8-9)
- Use proof by induction for n choose k to derive formula for k squared (Example #10a-b)
- Find the integer coefficients and formula for k^2 (Example #10c-d)
Recursive Formula
1 hr 49 min 25 Examples
- Can you guess the pattern and determine the next term in the sequence? (Examples #1-7)
- What is a Recursive Definition and Explicit Formula?
- Find the first five terms of the sequence (Examples #8-10)
- Recursive formula and closed formula for Arithmetic and Geometric Sequences
- Triangular — Square — Cube — Exponential — Factorial — Fibonacci Sequences
- Discover a recursive definition for each sequence (Examples #11-14)
- Use known sequences to find a closed formula (Examples #15-20)
- Using reverse—add method on Arithmetic Sequences (Examples #21-22)
- Summing Geometric Sequences using multiply—shift—subtract method (Examples #23-34)
- Summation and Product Notation (Example #25a-d)
Iteration Method Recurrence
1 hr 36 min 8 Examples
- Overview of how to solve a recurrence relation using backtracking iterations
- Use iteration to solve for the explicit formula (Examples #1-2)
- Use backward substitution to solve the recurrence relation (Examples #3-4)
- Solve the recurrence relation using iteration and known summations (Examples #5-6)
- Find the closed formula (Examples #7-8)
Recurrence Relation
1 hr 39 min 8 Examples
- Identifying a linear homogeneous recurrence relation and its degree (Example #1a-i)
- What is the characteristic root method and formulas for distinct—repeated—complex roots?
- Solve the degree 1 recurrence relation (Example #2)
- Identify the closed formula with distinct roots (Examples #3-5))
- Find the closed form with repeated roots (Examples #6-7)
- Uncover the explicit formula with complex roots (Example #8)
Chapter Test
1 hr 0 min 13 Practice Problems
- Use the counting principle (Problems #1-2)
- Use combinations without repetition (Problem #3)
- Use combinations with repetition (Problem #4)
- Use permutations (Problem #5)
- How many ways can you select a committee with equal members (Problem #6a)
- How many ways can you select a committee if two people are not both selected (Problem #6b)
- Prove using the pigeonhole principle (Problem #7)
- Identify the indicated coefficient (Problem #8)
- Use the Binomial Expansion (Problem #9)
- Give the explicit formula using partial sums (Problem #10)
- Find the recursive formula (Problem #11)
- Solve the recurrence relation (Problem #12)
- Find the 40th term of the sequence (Problem #13)
