Factorial, Permutations, Combinations Week 6 TEST # 2 – Next week!

Slides:

Advertisements

Similar presentations

Counting Principles Probability.

Advertisements

In this lesson we single out two important special cases of the Fundamental Counting Principle permutations and combinations. Goal: Identity when to use.

Permutations Examples 1. How many different starting rotations could you make with 6 volleyball players? (Positioning matters in a rotation.)

Basic counting principles, day 1 To decide how many ways something can occur, you can draw a tree diagram. Note that this only shows half of the tree –

Counting Principles The Fundamental Counting Principle: If one event can occur m ways and another can occur n ways, then the number of ways the events.

5.4 Counting Methods Objectives: By the end of this section, I will be able to… 1) Apply the Multiplication Rule for Counting to solve certain counting.

___ ___ ___ ___ ___ ___ ___ ___ ___

12.1 & 12.2: The Fundamental Counting Principal, Permutations, & Combinations.

CSE115/ENGR160 Discrete Mathematics 04/17/12

CSRU1100 Counting. Counting What is Counting “how many?” Counting is the process of determining the answer to a question of “how many?” for any given.

Chapter 2 Section 2.4 Permutations and Combinations.

MATH104 Ch. 11: Probability Theory

College Algebra Fifth Edition

Permutations and Combinations. Random Things to Know.

3.4 Counting Principles Statistics Mrs. Spitz Fall 2008.

DAY 2 OF CHAPTER 13 NOW LET’S THINK ABOUT FAMILIES WITH 3 CHILDREN A) CREATE A SAMPLE SPACE OF EQUALLY LIKELY OUTCOMES B) WHAT’S THE PROBABILITY A 3-CHILD.

Advanced Math Topics Chapters 4 and 5 Review. 1) A family plans to have 3 children. What is the probability that there will be at least 2 girls? (assume.

Conditional Probabilities Multiplication Rule Independence.

Factorials How can we arrange 5 students in a line to go to lunch today? _________ __________ __________ __________ ________.

Do Now: Make a tree diagram that shows the number of different objects that can be created. T-shirts: Sizes: S, M, L and T-shirts: Sizes: S, M, L and Type:

Review of 5.1, 5.3 and new Section 5.5: Generalized Permutations and Combinations.

Permutations and Combinations Multiplication counting principle: This is used to determine the number of POSSIBLE OUTCOMES when there is more than one.

Transparency 3 Click the mouse button or press the Space Bar to display the answers.

Permutations Lesson 10 – 4. Vocabulary Permutation – an arrangement or listing of objects in which order is important. Example: the first three classes.

Chapter 4 Lecture 4 Section: 4.7. Counting Fundamental Rule of Counting: If an event occurs m ways and if a different event occurs n ways, then the events.

MATH104 Ch. 11: Probability Theory. Permutation Examples 1. If there are 4 people in the math club (Anne, Bob, Cindy, Dave), and we wish to elect a president.

Permutations Chapter 12 Section 7. Your mom has an iTunes account. You know iTunes requires her to enter a 7- letter password. If her password is random,

Methods of Counting Outcomes BUSA 2100, Section 4.1.

Section 2.6: Probability and Expectation Practice HW (not to hand in) From Barr Text p. 130 # 1, 2, 4-12.

Permutations, Combinations, and Counting Theory AII.12 The student will compute and distinguish between permutations and combinations and use technology.

Lesson 9-4 Pages Permutations Lesson Check 9-3.

Quiz 10-1, Which of these are an example of a “descrete” set of data? 2.Make a “tree diagram” showing all the ways the letters ‘x’, ‘y’, and ‘z’

Prob/Stats Definition A permutation is an ordered arrangement of objects. (For example, consider the permutations of the letters A, B, C and D.)

Warm Up 2/1/11 1.What is the probability of drawing three queens in a row without replacement? (Set up only) 2.How many 3 letter permutations can be made.

10/23/ Combinations. 10/23/ Combinations Remember that Permutations told us how many different ways we could choose r items from a group.

Simple Arrangements & Selections. Combinations & Permutations A permutation of n distinct objects is an arrangement, or ordering, of the n objects. An.

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 1 M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY S TATISTICS Section 3-6 Counting.

Sullivan Algebra and Trigonometry: Section 14.2 Objectives of this Section Solve Counting Problems Using the Multiplication Principle Solve Counting Problems.

Quiz Plot the point: (-4, 2, -3) in the Cartesian space. Find the midpoint between the 2 points: P(1, 5, -7) and Q(-5, 3, -3) 3. Find the distance.

2.2 Permutations (Textbook Section 4.6). Recall Factorial Notation  The number of ways to arrange n objects into n spots is n! (read “!” a “factorial”)

Permutations, Combinations, and Counting Theory

Counting Principles Multiplication rule Permutations Combinations.

Algebra 2/TrigonometryName: __________________________ 12.1, 12.2 Counting Principles NotesDate: ___________________________ Example 1: You are buying.

6.7 Permutations & Combinations. Factorial: 4! = 4*3*2*1 On calculator: math ==> PRB ==> 4 7! = 5040 Try 12!

11.1A Fundamental Counting Principal and Factorial Notation 11.1A Fundamental Counting Principal If a task is made up of multiple operations (activities.

What is a permutation? A permutation is when you take a group of objects or symbols and rearrange them into different orders Examples: Four friends get.

2.1 Factorial Notation (Textbook Section 4.6). Warm – Up Question  How many four-digit numbers can be made using the numbers 1, 2, 3, & 4?  (all numbers.

HAWKES LEARNING Students Count. Success Matters. Copyright © 2015 by Hawkes Learning/Quant Systems, Inc. All rights reserved. Section 7.2 Counting Our.

Algebra-2 Counting and Probability. Quiz 10-1, Which of these are an example of a “descrete” set of data? 2.Make a “tree diagram” showing all.

Permutations and Combinations PSSA Unit. Permutations A permutation of the letters abc is all of their possible arrangements: abc acb bac bca cab cba.

MATH260 Ch. 5: Probability Theory part 4 Counting: Multiplication, Permutations, Combinations.

Copyright © Peter Cappello 2011 Simple Arrangements & Selections.

Multiplication Counting Principle How many ways can you make an outfit out of 2 shirts and 4 pants? If there are m choices for step 1 and n choices for.

Special Topics. Calculating Outcomes for Equally Likely Events If a random phenomenon has equally likely outcomes, then the probability of event A is:

PreQuiz 12A Make a systematic list of all possible outcomes: You spin a spinner and flip a coin.

Quiz: Draw the unit circle: Include: (1)All “nice” angles in degrees (2) All “nice” angles in radians (3) The (x, y) pairs for each point on the unit circle.

0.4 Counting Techniques. Fundamental Counting Principle TWO EVENTS:If one event can occur in m ways and another event can occur in n ways, then the number.

Permutations and Combinations. Fundamental Counting Principle Fundamental Counting Principle states that if an event has m possible outcomes and another.

Counting Techniques Section 5.5. Objectives Solve counting problems using the Multiplication Rule Solve counting problems using permutations Solve counting.

Multiplication Rule Combinations Permutations

CSE15 Discrete Mathematics 04/19/17

combinaTorial Analysis

12-5 Combinations.

Permutations and Combinations

Counting, Permutations, & Combinations

6-7 Permutations and Combinations

Combinations & Permutations (4.3.1)

Probability Warm Up page 12- write the question you have 10 mins to complete it. See coaching on page 85.

Permutations and Combinations

Presentation transcript:

Factorial, Permutations, Combinations Week 6 TEST # 2 – Next week! Probability Factorial, Permutations, Combinations Week 6 TEST # 2 – Next week!

Permutations are arrangements of n (number of objects) in a specific order. With permutations “order matters”! Problems involve the words: order, different, arrange, specific, place, position, rank, anything that has to do with a specific spot.

Factorial Notation Formula: n! = n*(n-1)*(n-2)*…..*(1) Is a shorthand way to express multiplication of decreasing, consecutive integers. Formula: n! = n*(n-1)*(n-2)*…..*(1) (Ex. 1) 5! = 5*4*3*2*1 = 120. (Ex. 2) 0! = 1 (Ex. 3) 9! =

Uses for Factorial: How many different ways can I arrange the letters in my first name: JOE ? JOE, JEO, OJE, OEJ, EJO, EOJ = 6ways There are three letters, thus, 3! 3! = 3*2*1 = 6 ways.

More Examples: (Ex. 1) How many ways can a coach arrange a line-up of 6 baseball players? (Ex. 2) In how many different ways can I re-arrange the seating of 8 people? (Ex. 3) How many different ways can I arrange 10 questions on a quiz?

More Examples: (Ex. 4) How many different ways can I arrange the letters in the word MATH? (Ex. 5) How many different ways can I arrange the letters in the word PASS? (Ex. 6) How many different ways can I arrange the letters in the word TEXTBOOK? (Ex. 7) How many different ways can I arrange the letters in the word STATISTICS?

What about: MISSISSIPPI?

Smaller Arrangements of Larger Group Permutation Rule – is the arrangement of n objects in a specific order, using only r at a time. The notation for a permutation: n Pr n is the total number of objects r is the number of objects chosen (want)

n Pr Formula: n Pr = n!/(n-r)! (ex 1) 6 P4 = 6!/(6-4)! = 6!/2! = (6*5*4*3*2!)/2! = = 360 (ex 2) 8 P3 = (ex 3) 5 P5 = (ex 4) How many different 3-digit numerals can be made from the digits 4, 5, 6, 7, 8 if a digit can appear just once in a numeral? 5 P3 = 5·4·3 = 60

Sabres Line-up: In how many ways can Lindy Ruff arrange 13 forwards in front lines of 3 players? (Order matters here because there is a center, a right wing, and a left wing.) 13 P3 = 1,716 ways

Statistics Prize Money! It’s time for the big pay-out! The college is going to pay-out three places to students in Thursday night Statistics. 1st = $5,000; 2nd = $3,000; 3rd = $1,000 With 26 students in the class, find the following: a) P(1st Place) = b) P(Winning) = c) How many different arrangements of winners can be made from our class?

Special Arrangements: How many different license plates can NY State issue if they are to have 3 letters followed by 4 numbers? How many different license plates can NY State issue if they are to have 3 different letters followed by 4 different numbers?

Special Arrangements: A new area code is being created. How many phone numbers are being created if the following specifications are met? The 1st number cannot be a ZERO or a ONE The first three cannot be 911 or 411.

Combinations Combination: A set of objects in which position (or order) is NOT important. How many different groups of 3 can be formed including Deb, Lydia, and Jessica? (3 people) In a combination, the trio of Deb, Lydia, and Jessica is THE SAME as Jessica, Lydia, and Deb. Thus, there is only one group.

n Cr Formula: n Cr = n!/(r!(n-r)!) (ex 1) 6 C4 = 15 (ex 2) 9 C3 = (ex 4) How many different 3-letter combo’s can be made from the letters A, B, C, D, E if a letter can appear just once in a combo? 5 C3 = 10

Permutation versus Combination What’s the diff’? Permutation versus Combination 1. Picking a team captain, pitcher, and shortstop from a group. 1. Picking three team members from a group. 2. Picking your favorite two colors, in order, from a color brochure. 2. Picking two colors from a color brochure. 3. Picking first, second and third place winners. 3. Picking three winners.

Prize Winners (Ex 1) – A raffle has 20 entries. The prizes include 5 gift certificates, all for $20 each. How many different groups can be selected to claim the prizes? (Ex 2) – 15 people placed their names in a hat to win trip to beautiful downtown Sanborn. If the prize commission is only choosing 8 winners, how many groups can be formed?

Sabres Line-up: In how many groups can Lindy Ruff arrange 13 forwards in front lines of 3 players? (If order does not matter, each player could change up to be a center, a right wing, and a left wing.) 13 C3 =

Form a committee… (Ex 1) – A committee is to be formed consisting of 3 people. There are 5 people to choose from, how many different committees can be created? (Ex 2) – A committee is to be formed consisting of 5 people. There are 12 people to choose from, how many different committees can be created?

Special Emergency Committee A new committee is to be formed from a group of 20 college students. It is to have 6 members and it must contain an equal amount of boys as girls. There are 12 boys in the original group. How many groups can be formed?