Math 14 Lecture Notes Ch. 3.6

Transcription

1 Math Lecture Notes h... ounting Rules xample : Suppose a lottery game designer wants to list all possible outcomes of the following sequences of events: a. tossing a coin once and rolling a -sided die once omplete the tree diagram branch labels and list the outcomes in the sample space. Outcomes Number of outcomes of tossing a coin: Number of outcomes of rolling a -sided die: Number of outcomes in this sample space: = b. tossing a coin times. Outcomes Number of outcomes of tossing a coin: Number of outcomes of tossing a coin: Number of outcomes of tossing a coin: Number of outcomes in this sample space: = Page of

2 Math Lecture Notes h.. To find certain probabilities, we don t need a list of all outcomes in the event and sample spaces, but we do need to know the number of outcomes in each. The examples on the previous page illustrate the following principle. undamental ounting Principle In a sequence of n events in which the first one has k possibilities and the second event has k possibilities, and so forth, the total number of possibilities of the sequence will be k k k! k n xample : Use the fundamental counting principle to determine the number of student classifications according to gender (male, female), academic level (high school, freshman, sophomore), and enrollment in statistics (enrolled, not enrolled). ender male female cademic level high school freshman sophomore nrollment in statistics enrolled not enrolled = xample : Use the fundamental counting principle to determine the number of license plates that can be made using one digit followed by letters, followed by digits st nd rd th th th th N O P Q R S T H U I V J W K X L Y M Z N O P Q R S T H U I V J W K X L Y M Z N O P Q R S T H U I V J W K X L Y M Z = xample : Use the fundamental counting principle to determine the number of distinct codons that can be formed from the nucleotides,, U,, and, taken at a time. Page of

3 Math Lecture Notes h.. Page of xample : Use the fundamental counting principle to determine the number of cell phone customers that can be served if the area code and first three digits must be or () xxxx or () - xxxx Total st nd rd th st nd rd th + = xample : the number of -digit I numbers that can be made if no digit can be repeated st nd rd th th (take (take (take (take = xample : the number of -digit I numbers if no digit can be repeated st nd rd th th th th th th th (take (take (take (take (take (take (take (take (take =

4 Math Lecture Notes h.. Suppose we had a set of distinct symbols and wanted to find the number of I numbers that could be made if no repeats are allowed. We would need to calculate!. It is cumbersome to write this in its entirety. Thus, mathematicians have created a shorthand called factorial notation. We write! =! (! is read as factorial) actorial notation will be used extensively in our work in permutations and combinations. Note: We define! =. The previous problem of finding the number of -digit I numbers for which no repeats are allowed is an example of a permutation. It is the number of possible arrangements of digits in a specific order. g In general, a permutation is an arrangement of n objects in a specific order. xample : Suppose a rancher has five horse and five stalls. How many ways can the rancher place the horses into the stalls? Let s call the horses,,,, and. st stall nd stall rd stall th stall th stall (take (take (take (take = There are! = = ways to place the horses. Page of

5 Math Lecture Notes h.. xample : Suppose there are only stalls. How many ways can we choose from the horses to place them into stalls? st stall nd stall rd stall (take (take = xample : How many ways can we choose horses from to place them into stalls? (We will continue to name them using consecutive letters.) st stall nd stall rd stall H I J H I J (take H I J (take = xample : How many ways can we choose horses from to place them in stalls? st stall nd stall rd stall th stall th stall th stall (take (take (take (take (take = Page of

6 Math Lecture Notes h.. Notice that = =!! =! ( )! and = =!! =! ( )! and = =!! =! ( )!. We can use this emerging pattern to write a formula for finding arrangements of n objects taken r at a time. Permutation Rule The arrangement of n objects in a specific order using r objects at a time is called a permutation of n objects taken r objects at a time. It is written as n P r, and the formula is n! np r = (n r)! xample : Use the permutation rule to determine the number of distinct permutations possible of a padlock in which numbers from, no repetitions, that can be used to form the code to unlock the padlock. Page of

7 Math Lecture Notes h.. xample : Let's revisit our horses. Suppose we want to select of them to turn out into a pasture. How many ways can we select of the horses for this purpose? If we were placing them in stalls, we would use the formula! P = ( )! =! = = =.! However, the formula assumes that the order in which the selection is made matters. Let's list the permutations. In choosing of the horses to turn out into the pasture, we count the same as, the same as, the same as, and so on. Thus, there are = ways to choose from horses, at a time without regard to order. We call this kind of selection a combination. g combination is a selection of objects without regard to order. Let's develop a formula for finding combinations of n objects selected r at a time. xample : How many ways can we choose horses at a time without regard to order? There are P = permutations of choosing horses at a time if order matters. Since order does not matter, we divide out the duplicates. ut how many duplicates do we have when choosing at a time? Here is a list of the permutations. Notice that the arrangements in each column all represent the permutations of objects! taken at a time. ach column contains P = ( )! =!! =! =! To find the number of ways to choose horses at a time without regard to order, we should divide by what number? Page of

8 Math Lecture Notes h.. xample : How many ways can we choose horses at a time without regard to order? There are P = permutations of choosing horses at a time if order matters. Order does not matter, so we divide out the duplicates, P = r!. Let s develop a formula now. We will be using the notation n r to represent the number of combinations of choosing n objects r at a time. We have found that n r = n P r r! = n! (n r)! r!= n! (n r)!r! ombination Rule The number of selections possible of r objects chosen from n objects is called a combination of n objects taken r objects at a time. It is written as n r, and the formula is n! n r = (n r)!r! xample : Use the combination rule to determine how many ways candies can be selected from a dish of candies if order is to be disregarded. xample : Use the combination rule to determine how many ways can a jury of women and men be selected from a pool of women and men? Page of

9 Math Lecture Notes h.. ombinations and Permutations What's the ifference? In nglish we use the word "combination" loosely, without thinking if the order of things is important. In other words: "My fruit salad is a combination of apples, grapes and bananas" We don't care what order the fruits are in, they could also be "bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad. "The combination to the safe was ". Now we do care about the order. "" would not work, nor would "". It has to be exactly --. So, in Mathematics we use more precise language: If the order doesn't matter, it is a ombination. If the order does matter it is a Permutation. In other words: Permutation is an ordered ombination. Label each as Permutation (P) or ombination () problems. a. How many ways can we choose 's from a stack of 's? b. How many ways can books be arranged on a shelf? c. How many ways can of dogs be placed in kennel cages? d. How many ways can of dogs be turned out into a play yard? e. How many ways can of flowers be given to people? f. g. How many ways can candidates be chosen from candidates for final interviews? How many license plates can be made using letters followed by numbers? h. How many ways can radio commercials be run during an hour Page of

10 Math Lecture Notes h.. emonstration Problems valuate each expression.. (a)! Practice Problems valuate each expression.. (b)!. (a)!!. (b)!!. (a) P. (b) P. (a) P. (b) P. (a). (b) nswers:. (b) ;. (b) ;. (b) ;. (b) ;. (b) Page of

11 Math Lecture Notes h.. emonstration Problems. (a) How many different I card numbers can be made if there are digits? Practice Problems. (b) How many different I card numbers can be made if there are digits?. (a) How many license plates can be made if the first characters are letters and the next are digits?. (b) How many license plates can be made if the first characters are letters and the next are digits?. (a) How many different I card numbers can be made if there are digits and no digit can be used more than once?. (b) How many different I card numbers can be made if there are digits and no digits can be used more than once?. (a) How many ways can dogs be chosen from dogs to be placed into kennels?. (b) How many ways can dogs be chosen from dogs to be placed into kennels?. (a) How many ways can dogs in a shelter be chosen to be turned out into the play yard at a time?. (b) How many ways can dogs in a shelter be chosen to be turned out into the play yard at a time? nswers:. (b),;. (b),,;. (b),;. (b),;. (b) Visit to learn more about the history and future of license plates. Page of