Discrete Mathematics
WEN-CHING LIEN Department of Mathematics National Cheng Kung University
2008
1.1: The Rules of Sum and Product
Theorem (The Rule of Sum)
If a first task can be performed in m ways ,while a second task can be performed in n ways , and the two tasks cannot be performed simultaneously , then performing either task can be accomplished in any one of m+n ways.
Example (1.1)
A college library has 40 textbooks on sociology and 50 textbooks dealing with anthropology.
By the rule of sum , a student at this college can select among 40+50=90 textbooks in order to learn more about one or the other of these two subjects.
1.1: The Rules of Sum and Product
Theorem (The Rule of Sum)
If a first task can be performed in m ways ,while a second task can be performed in n ways , and the two tasks cannot be performed simultaneously , then performing either task can be accomplished in any one of m+n ways.
Example (1.1)
A college library has 40 textbooks on sociology and 50 textbooks dealing with anthropology.
By the rule of sum , a student at this college can select among 40+50=90 textbooks in order to learn more about one or the other of these two subjects.
Theorem (The Rule of Product)
If a procedure can be broken down into first and second stages,and if there are m possible outcomes for the first stage and if,for each of these outcomes,there are n possible
outcomes for the second stage ,then the total procedure can be carried out, in the designated order,in mn ways.
Example (1.5)
The drama club of Central University is holding tryouts for a spring play. with six men and eight women auditioning for the leading male and female roles,by the rule of product the director can cast his leading couple 6×8=48 ways.
Theorem (The Rule of Product)
If a procedure can be broken down into first and second stages,and if there are m possible outcomes for the first stage and if,for each of these outcomes,there are n possible
outcomes for the second stage ,then the total procedure can be carried out, in the designated order,in mn ways.
Example (1.5)
The drama club of Central University is holding tryouts for a spring play. with six men and eight women auditioning for the leading male and female roles,by the rule of product the director can cast his leading couple 6×8=48 ways.
Example 1.8–
At times it is necessary to combine several different counting principles in the solution of one problem. Here we find that the rules of both sum and product are needed to attain the answer.
At the AWL corporation Mrs. Foster operates the Quick snack Coffee Shop. The menu at her shop is limited: six kinds of muffins, eight kinds of sandwiches, and five beverages(hot coffee,hot tea,iced tea,cola,and orange juice ).Ms.Dodd,an editor at AWL,sends her assistant Carl to the shop to get her lunch–either a muffin and a hot beverage or a sandwich and a cold beverage.
By the rule of product,there are 6×2=12ways in which Carl can purchase a muffin and hot beverage.A second applicaton of this rule shows that there are 8×3=24 possibilities for a sandwich and cold beverage.So by the rule of sum,there are 12+24=36 ways in which Carl can purchase Ms.Dodd’s
Example 1.8–
At times it is necessary to combine several different counting principles in the solution of one problem. Here we find that the rules of both sum and product are needed to attain the answer.
At the AWL corporation Mrs. Foster operates the Quick snack Coffee Shop. The menu at her shop is limited: six kinds of muffins, eight kinds of sandwiches, and five beverages(hot coffee,hot tea,iced tea,cola,and orange juice ).Ms.Dodd,an editor at AWL,sends her assistant Carl to the shop to get her lunch–either a muffin and a hot beverage or a sandwich and a cold beverage.
By the rule of product,there are 6×2=12ways in which Carl can purchase a muffin and hot beverage.A second applicaton of this rule shows that there are 8×3=24 possibilities for a sandwich and cold beverage.So by the rule of sum,there are 12+24=36 ways in which Carl can purchase Ms.Dodd’s
Example 1.8–
At times it is necessary to combine several different counting principles in the solution of one problem. Here we find that the rules of both sum and product are needed to attain the answer.
At the AWL corporation Mrs. Foster operates the Quick snack Coffee Shop. The menu at her shop is limited: six kinds of muffins, eight kinds of sandwiches, and five beverages(hot coffee,hot tea,iced tea,cola,and orange juice ).Ms.Dodd,an editor at AWL,sends her assistant Carl to the shop to get her lunch–either a muffin and a hot beverage or a sandwich and a cold beverage.
By the rule of product,there are 6×2=12ways in which Carl can purchase a muffin and hot beverage.A second applicaton of this rule shows that there are 8×3=24 possibilities for a sandwich and cold beverage.So by the rule of sum,there are 12+24=36 ways in which Carl can purchase Ms.Dodd’s