Download Fundamental Counting Principle (the multiplication principle)

Pre-­‐Calculus Mathematics 12 – 7.1 – Combinatorics Goals: 1. Identify the Fundamental Counting Principle 2. Use Tree Diagrams to solve counting problems 3. Introduce and apply Factorial Notation Combinatorics involves a systematic way of counting. These systems can be used to solve complex counting problem such as the study of objects arrangements and their probabilities. There are new principles, notations, and language required throughout the study of combinatorics. Fundamental Counting Principle (the multiplication principle) For a task made up of options, the total number of possibilities = a ·∙ b ·∙ c ·∙ .... where a = the number of choices for the 1st option b = the number of choices for the 2nd option c = the number of choices for the 3rd option, ... etc. Example 1: A restaurant menu offers a meal that consists of an appetizer, main dish and dessert. If there are 3 appetizer options, 4 main dish options, and 2 dessert options, how many possible meals are there? Example 2: A truck license plate in BC consists of 6 characters. The first 2 characters can be any of the letters. The next 4 characters can be any of the digits from 0 to 9. How many license plates are possible? Example 3: Assume a car license plate consists of 7 characters. The first 3 characters can be any of the letters from A to F, but no letter can be repeated. The next 3 characters can be any of the digits from 1 to 9, but no digit can be repeated. The last character can be any of the letters X, Y or Z. An example of this format is: BFA648Y. How many license plates are possible? Pre-­‐Calculus Mathematics 12 – 7.1 – Combinatorics Tree Diagrams A tree diagram is good way to visualize and count the number of outcomes for an event. Often, a tree diagram or the fundamental counting principle can be used to solve a problem. Example 4: Use a tree diagram to solve: A restaurant menu offers a meal that consists of an appetizer, main dish and dessert. If there are 3 appetizer options, 4 main dish options, and 2 dessert options, how many possible meals are there? If a counting problem is more complex, a tree diagram can be an easier method to finding the solution. Example 5: Kristin and Christine are in a fishing derby. The first person to catch 2 fish in a row or a total of three fish wins. How many outcomes are possible? Practice: Page 319 #1 -­‐ 15 Pre-­‐Calculus Mathematics 12 – 7.1 – Combinatorics Factorial Notation Consider how many ways there are to arranging 6 different books side by side on a shelf. The fundamental counting principle can be used to calculate this as 6 x 5 x 4 x 3 x 2 x 1. We can write this in a simplified form using factorial notation. 6 x 5 x 4 x 3 x 2 x 1 = Definition of n factorial n! = n n − 1 n − 2 n − 3 n − 4 … .3 ∙ 2 ∙ 1 where n is a positive integer. 0! = 1 by definition Example 6: Without using a calculator, evaluate: a) !"!
!"!
b) !"!∙!"!
!"!∙!"!
Example 7: Simplify. (!!!)!
!!! !
Pre-­‐Calculus Mathematics 12 – 7.1 – Combinatorics Example 8: Simplify. !!! !! ! !!! !
!! ! ! !!! !
Example 9: Solve: (!!!)!
!!
= 42 Practice: Page 319 #16 -­‐ 19 Pre-­‐Calculus Mathematics 12 – 7.2 – Permutations Goals: 1. Solve permutation problems with distinguishable objects 2. Solve permutation problems with indistinguishable objects Permutation A permutation is an arrangement of a set of elements in which the order of the elements is important. We will look at permutations that have distinguishable objects (all different) and indistinguishable objects (some are the same). A permutation is just the fundamental counting principle put into a formula. Distinguishable objects all arranged The number of possible permutations (order arrangements) of “n” distinct objects is n! Example 1: Six friends go to watch a movie and all sit together. How many ways can this occur? Example 2: Six friends go to watch a movie and all sit together. How many ways can this occur done if Tabitha must sit next to John? Example 3: Six friends go to watch a movie and all sit together. How many ways can this occur done if Tabitha must not sit next to John? Pre-­‐Calculus Mathematics 12 – 7.2 – Permutations Permutations where not all objects are arranged The number of possible arrangements of “n” distinct objects taken “r” at a time is n Pr =
P n, r =
!!
!!! !
(n objects – only r ordered spaces available) Example 4: How many 4 digit combinations can be created using the numbers 0 – 9 with no repeats? Example 5: How many different ways can a president, vice-­‐president, and secretary be selected from a class of 18? Example 6: Determine the number of five letter “words” that can be formed from the letters of the word FACTORED. Example 7: Solve for n in nP2 = 90 Practice: Page 326 #1 -­‐ 11 Pre-­‐Calculus Mathematics 12 – 7.2 – Permutations Permutations with indistinguishable objects The number of permutations of “n” objects where a objects are alike, another b objects are alike, another c objects are alike,......etc. 𝐧!
Permutations = 𝐚!𝐛!𝐜!
Example 7: There are 12 cars of the same model in a car dealership. If 6 are blue, 4 are green and 2 are red, how many ways can the cars be arranged? Example 8: a) How many different ways can the letters of the word INTENTIONAL be arranged? b) How many of these arrangements begin with the letter T? c) How many of these arrangements have all the N’s together? Practice: Page 326 #12 -­‐ 19 Pre-­‐Calculus Mathematics 12 – 7.3 – Combination Goal: Solve combination problems with distinguishable objects Warm-­‐up: Alan, Brent, Cole, and Dave are running a race. First place gets $50, second place gets $25 and third place gets $10. List all the possible outcomes of the race and determine the number of ways that the prizes can be awarded. If the top three get the same prize, how many ways the prizes can be awarded. Combination A combination is an arrangement of a set of elements in which the order of the elements is not important. The number of combinations (in any order) of “n” distinct objects taken “r “at a time is given by 𝑛
= C n, r =
!𝐶! =
𝑟
! !!
!!
= !!
!!! !
!!
=
!!
!!! !!!
; n, r > 0 Example 1: A hockey team of 18 players is chosen from 25 available players. How many ways can this occur? Example 2: To win Lotto 6/49, a person must select six numbers between 1 and 49. How many different selections of numbers are possible? Pre-­‐Calculus Mathematics 12 – 7.3 – Combination Example 3: How many committees of 5 teachers can be selected from 7 male teachers and 5 female teachers if the committee must contain 2 female teachers? Example 4: Poker is a card game played from a deck of 52 cards. a) How many different five card poker hands are possible? b) In how many of the hands in a) will there be: i) all diamonds? ii) 4 black cards and 1 red card iii) 3 kings and 2 queens iv) 3 kings? v) four aces? vi) 5 cards of the same suit? Example 5: Consider a standard deck of 52 cards. How many five card hands can occur that contain at least three diamonds? Practice: Page 331 #1 -­‐ 11 Pre-­‐Calculus Mathematics 12 – 7.4 – Binomial Theorem Goal: 1. Expand using the Binomial expansion theorem 2. Find a specific term in a binomial expansion Blaise Pascal, a French mathematician, described a triangular array of numbers corresponding to the number of ways to choose r elements from a set of n objects. Each number in Pascal’s triangle can be written in the notation n C r where n is the number of objects in the set and r refers to the number selected. We refer to this array now as Pascal’s Triangle. We can now use this to complete the following table: Following the pattern above, expand (x + y)4 with coefficients in n C r form Binomial Expansion The binomial expansion theorem is a method of expanding an expression of the form a + b ! , where n is a positive integer. !
a + b
!
nCr ∙ a!!! ∙ b! =
!!!
Pre-­‐Calculus Mathematics 12 – 7.4 – Binomial Theorem Example 1: Expand. x + y ! Example 2: Expand. 3x − 2y ! ! !
Example 3: Expand. 4x ! − ! Term Formula Often, specific terms of an expansion are need and rather then expand the entire binomial; we can use part of the formula to find the specific term. t !!! = nCr ∙ a!!! ∙ b! Example 4: Determine the sixth term in the expansion of 2x − 3 ! Pre-­‐Calculus Mathematics 12 – 7.4 – Binomial Theorem Example 5: Determine the 23rd term in the of a + b !" = Example 6: Which term of x ! −
! !
!
is a constant? Example 7: Determine b, such that x + b ! has a term –13 608x3. Practice: Page 336 # 1 – 6, 9 -­‐ 14 Pre-­‐Calculus Mathematics 12 – 7.5 – Pathway Problems Goal: Solve pathway problems using permutations and Pascal’s Triangle Pathway problems are a type of counting situation used in networking problems. The simplest form of pathway is when there are only two options for directions. Consider the situation: Count the number of different pathways to get from A to B moving only south and/or east. Rather than counting, we can solve questions like these using two methods. Method 1: Permutations: Method 2: Pascal’s Triangle: Example 1: Find the number of different pathways to get from A to B moving only south and/or east using two different methods. Pre-­‐Calculus Mathematics 12 – 7.5 – Pathway Problems Example 2: Find the number of different pathways to get from A to B moving only south and/or east. Example 3: Determine the number of different pathways from the game piece marked O to the square marked W on the other side of the board if the pieces can only move forward and diagonally. Practice: Page 339 # 1 + 2