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12.1 – The Fundamental Counting Principle and Permutations The Fundamental Counting Principle Two Events: If one event can occur in ways and another event can occur in ways then the number of ways both events can occur is Three or More Events: The fundamental counting principle can be extended to three or more events. If three events can occur in and ways, then the number of ways all events can occur is Examples: 1. At a restaurant, you have the choice between 8 different entrées , 2 different salads, 12 different drinks, and 8 different desserts. How many different dinners consisting of 1 entrée, 1 salad, 1 drink and 1 dessert can you choose? 2. How many different 7 digit phone numbers are possible if the first digit can’t be 0 or 1? 3. The standard configuration of a NY license plate is 3 digits followed by 3 letters. a. How many different license plates are possible if digits and letters can be repeated? b. How many different license plates are possible if digits and letters can’t be repeated? An ordering of objects is a permutation of the objects. For example, there are six permutations of the letters A, B & C: ABC, ACB, BAC, BCA, CAB, CBA We can use the fundamental counting principle to determine the # of permutations of objects: ( ) In general, the number of permutations of n distinct objects is ( ) We say factorial. Examples: 1. You have homework assignments from 5 different classes to complete this weekend. a. In how many different ways can you complete the assignments? b. In how many different ways can you choose 2 of the assignments to complete first and last? 2. There are 8 movies you would like to see currently showing in theaters. a. In how many different ways can you see all 8 movies? b. In how many different ways can you choose a movie to see Saturday and a movie to see Sunday? The number of permutations of objects taken from a group of is denoted by distinct objects ( ) There is a calculator key to do this on the graphing calculator. Example: You are considering 10 different colleges. Before you decide to apply to the colleges, you decide you want to visit all or some of them. In how many different orders can you visit a. 6 of them? ( ) You are looking for the number of b. all of them? permutations of 10 objects taken 6 at 10 objects taken 10 at a time a time. Try one: There are 12 books on the summer reading list. In how many orders can you read: a. 4 of them? b. all of them? Permutations with Repetition The number of distinguishable permutations of n objects where one object is repeated time and another object is repeated times, and so on is Example: Find the number of distinguishable permutations of the letters in a. OHIO 4 letters O is repeated 2 times b. MISSISSIPPI Homework: page 705-706 15 – 60 multiples of 5 12.2 – Combinations and The Binomial Theorem A combination is a selection of r objects from a group of n objects where order is not important. Combination of n objects taken r at a time: The number of combinations of r objects taken from a group of n distinct objects is denoted by: ( ) There is a calculator key to do this on the graphing calculator. Examples: A standard deck of 52 playing cards has 4 suits with 13 different cards in each suit. 1) If the order is which the cards are dealt is not important, how many different 5card hands are possible? ( ) 2) In how many of these hands are all five of the cards the same suit? You would need to choose 1 of the 4 suits and then 5 of the 13 cards in that suit. Try One – Use a standard deck of 52 cards. 1) If order is not important, how many different 7 card hands are possible? 2) How many of these hands are all 7 cards of the same suit? Some Examples: A restaurant serves omelets that can be ordered with any of the following ingredients: vegetarian : green peppers, red peppers, onion, mushroom, tomato, cheese meat: ham, bacon, sausage, steak a. Suppose you want exactly 2 vegetarian ingredients and 1 meat ingredient. How many different types of omelets can you order? 2 of 6 vegetable and 1 of 4 meat b. Suppose you can afford at most 3 ingredients in your omelet. How many different types of omelets can you order? Since you can have at most 3 ingredients this means you can choose an omelet with 0, 1, 2 or 3 ingredients. Because there are 10 items to choose from the number of possible omelets is: or you can do this in the calculator. A theater is staging a series of 12 different plays. You want to attend at least 3 of them. How many different combinations of plays can you attend? at least 3 means 3 plays or 4 plays or 5 plays……. … or 12 plays another way to look at it is there are 12 plays each of which we can attend or not attend = minus the ( ) Try some: 1. You are taking a vacation. You can visit as many as 5 different cities and 7 attractions. a) Suppose you want to visit exactly 3 different cities and 4 different attractions. How many different trips are there? b) Suppose you want to visit at least 8 locations (cities or attractions.) How many different types of trips are possible? 2. A restaurant offers 6 salad toppings. On a deluxe salad you can have up to 4 toppings. How many different combinations of toppings can you have? 3. How many possible 5-card hands contain exactly 3 kings? 4. From a group of 20 volunteers, you are choosing at least 18 to be peer counselors. In how many different ways could this be done? Homework: page 712-713: 18 – 30 multiples of 3, 48-54 even 12.3 – An Introduction to Probability The probability of an event is a number between 0 and 1 that indicates the likelihood the event will occur. A probability of 1 indicates that the event is certain to occur, while a probability of 0 indicates the event cannot occur. A probability of indicates that it is equally likely to occur or not occur. The Theoretical Probability of an Event: When all outcomes are equally likely, the theoretical probability that an event will occur is: ( ) The theoretical probability of an event is often simply called the probability of the event. Example: You are rolling a six-sided die whose sides are numbered from 1 to 6. Find the probability of: a) rolling a 5 ( ) b) rolling an even number ( ) c) rolling a number less than 7 ( ) You can express the probability as a fraction, decimal or percent. Try one: A spinner has 8 equal-size sectors numbered from 1 to 8. Find the probability of: a) spinning a 6 b) spinning a # > 5 Example: You put a CD that has 8 songs in your CD player. You set the player to play the songs at random. The player plays all 8 songs without repeating any song. a) What is the probability that the songs are played in the same order they are listed on the CD? ( ) b) You have 4 favorite songs on the CD. What is the probability that 2 of your favorite songs are played first, in any order? There are different combinations of 2 songs. Of these favorite songs so ( contain 2 of your ) Try some: 1) Five cards are drawn from a standard 52-card deck. What is the probability that the first 2 cards are red? 2) You made 15 of 21 free throw attempts. Find the probability that you will make your next free throw. Sometimes it is not possible or convenient to find the theoretical probability of an event. In such cases you may be able to calculate an experimental probability by performing an experiment, conducting a survey or looking at the history of the event. Some probabilities are found by calculating a ratio of 2 lengths, areas or volumes. These are called geometric probabilities. Example: Find the probability that a randomly thrown dart would hit the shaded portion of the target shown. 4 Another one: A store is open from 8 am – 8 pm. The manager works from 9 am – 4 pm. What is the probability that the manager is there during the same time as a customer who arrives randomly during the store’s hours of operation and stays for 15 minutes? Homework: page 719 – 721: 12-20 even, 30-36 even, 39 & 40 7.7 – Statistics and Statistical Graphs statistics – numerical values used to summarize and compare sets of data. The following measures of central tendency are three commonly used statistics 1. The mean or average of numbers is the sum of the numbers divided by n. The mean is denoted by ̅ which is read as “x bar” the mean is ̅ For the data 2. The median of numbers is the middle number when the numbers are written in order. (If n is even, the median is the mean of the two middle numbers.) 3. The mode of numbers is the number or numbers that occurs most frequently. There may be one mode, no mode or more than one mode. Measures of central tendency tell you what the center of the data is. Other commonly used statistics are called measures of dispersion. They tell you how spread out the data is. The range is the difference between the greatest and least data value. The standard deviation describes the typical difference or deviation between the mean and the data value. Standard Deviation of a Data Set: The standard deviation (read sigma) of √ ( ̅) ( is ̅) ( ̅) Data Sets: Period 3 Grades: 87, 73, 92, 67, 95, 84, 80, 100, 94, 63, 88, 76, 92, 100, 68, 93, 85, 93, 84, 96, 84, 88, 92, 58, 84 Period 7 Grades: 93, 84, 81, 83, 78, 81, 65, 72, 74, 89, 72, 81, 84, 93, 90, 83, 92, 75, 98, 91, 94, 93, 87, 100 Find the mean, median and mode of the data sets listed above. Period 3 Period 7 Find the range and standard deviation of the data sets. What does this tell you? Using statistical graphs: box-and-whisker plot – a graph that encloses the middle half of the data set in a “box” and uses “whiskers” to extend to the minimum and maximum data values. The median divides the data set into two halves. The lower quartile is the median of the lower half and the upper quartile is the median of the upper half. Steps to draw a box-and-whisker plot: 1. Order the data from least to greatest 2. Find the minimum and maximum values 3. Find the median 4. Find the lower and upper quartiles. 5. Plot these 5 numbers on a number line 6. Draw the box, whiskers and a line segment through the median. Make a box-and-whisker plot for the data sets A histogram is a bar graph where data values are grouped in intervals of equal width. The number of data values in each interval is called the frequency of the interval. You must make a frequency distribution, which shows the frequency of each interval. Make a frequency distribution of the data sets. Make a histogram of the data sets. Homework page 449 – 450: 10 – 30 multiples of 5