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Counting & Venn diagram:
Counting:
In some questions you are asked to count something: How many apples did Jorge buy, how many taka did Shelly spend, how many pages were read, how many ways are there to complete a job, etc. Some such problems may be solved using algebra. Others may need your to systematically make a list. Sometimes you may need to know some special strategies that we discussed.
Making a systematic list:
When a question asks 'how many', often the best strategy is to make a list of all the possibilities. You have to make this list systematically so that no item is left out or so that you don't count an item twice. Usually this means listing the possibilities in a numerical or alphabetical order. Inmost case, shortly, After starting to make the systematic list, you world see a pattern developing and
you should be able to find out how many elements should be in the list without having to write drown all of them.
Example: A palindrome is a number, such a 93539, that reads the same forward and backward. How many palindromes are there between 100 and 1000?
Solution: At first let us write the palindromes between 100 and 200.
101, 111, 121,131,141,151, 161, 171, 181, 191 = 10 Palindromes
Next comes the number between 200 and 300.
202, 212, 222, 232, 242, 252, 262, 272, 282, 292 =10 Palindromes
So, we can see a pattern. Each interval of 100 numbers contains 10 Palindromes. So, beginning with 1, 2, 3,.. 9 there are 9*10 = 90 Palindromes.
Answer: 90
This a good technique if the numbers in the problem are small. Proper use of this technique eliminates the risk of marking an error in arithmetic.
Using Arithmetic to count:
Let us go through a few examples. These will illustrate the basic rules of arithmetic we use for counting..
Example A: Bari entered the store with $213 & left with $ 203. How much did he spend in the store?
Example B: A man sitting in a cinema hall ticket counter sold ticket no. 213 through 209. How many tickets were sold?
Example C: Bari is the 213th person in a queue and Shaki is the 209th person. How many people are there between them?
Solution: (A-c): At first sight it may seem that all the three have the same answer 213-209=4. But actually, the answer for A is 4. B is 5 and C is 3.
In example A, Bari had $ 213 He spent $ no. 213, 212, 211, 210. He still had $ 2.9 left with him. So, he spend $4.
In example B. tickets numbered 213, 212, 211, 210, & 209 were sold . So, the total number is 5.
In the queue (example C) between Bari & Shakily there are person no. 212, 211 & 210 . So, the answer is 3.
In example A, we just subtracted, in example B we subtracted & then added a 1, in example C we subtracted & then subtracted 1 more.
The main point to note is whether or not the first and last numbers are included. In example A, Bari left the store with the $ 209. In example B. both the 213 & 209 ticket were sold. And in example C. neither the 213 person nor the 209 person was counted. So, ultimately we come to the following rules:
A) If exactly one of the end points is included, subtract.
B) If both the end points are included, subtract and add1 .
C) If none of the end points are included, subtract & subtract 1 more.
PDF] Math - Douglas Unified School District www.dusd.k12.az.us/LinkClick.aspx?fileticket=uvt6LFtK2Gw%3D...counting. Students create and use multiple representations such as charts, systematic lists, and tree diagrams. They note ... meals can be made from the menu? Make a systematic list of your possibilities. How can you .... Use the Venn diagram to determine if the ... of square units by cutting the triangular parts from the. Discrete Mathematics - Systematic Listing and Counting.htm www.ntuaft.com/.../CPI%20Checklists/.../Discrete%20Mathematics%20-...C. Discrete Mathematics—Systematic Listing and Counting ... Venn diagrams. 2. ... Building upon knowledge and skills gained in preceding grades, by the end of ... Explore counting problems involving Venn diagrams with three attributes (e.g., ...NJ ASK Grade 7 Mathematics Test and Item Specifications www.state.nj.us/njded/assessment/ms/5-7/specs/math7.htm C. Discrete Mathematics—Systematic Listing and Counting ... We have to make sure that the raised expectations for all students do not result in ... Numerical operations are an essential part of the mathematics curriculum, especially in the ..... C.2 Explore counting problems involving Venn diagrams with three attributes (e.g., ... SticiGui Counting www.stat.berkeley.edu/~stark/SticiGui/Text/counting.htm... Sampling Distributions · Scatterplots · Student's t Distribution · Venn Diagram (2 subsets) ... This chapter presents techniques and shortcuts for counting that are ... to approach counting systematically: Enumerating all the possibilities becomes very ... Counting large sets by listing all the possibilities is ..... I'll make you rich!".Sets and Venn Diagrams - the Australian Mathematical Sciences ...www.amsi.org.au/teacher_modules/Sets_and_venn_diagrams.htmlSuch counting quickly throws up situations that may at first seem contradictory. ... The Venn diagram makes the situation easy to visualise. ... The five elements of the set are separated by commas, and the list is enclosed .... b: If the set S in each part is finite, write down | S |. ..... working systematically around the diagram. Counting Methods (with Videos) - Online Math Learning www.onlinemathlearning.com/counting-methods.html Sets and Lists, Venn Diagrams, Inclusion-Exclusion Principle, Multiplication ... Uncertainty is part of the process of making decisions and predicting outcomes. ... we now turn to a discussion of counting objects in a systematic way before ... [PDF] Probability and Counting Rules B&W CONFIRMINGS highered.mcgraw-hill.com/sites/dl/free/007353496x/366095/Ch_04.pdf time a person awakes until he or she goes to bed, that person makes decisions regarding the possible events that are governed at least in part by chance. ... explains how the counting rules and the probability rules can be used .... systematically. .... The Venn diagram that represents the probability of the complement of an ...[PDF] Matwww.bedminsterschool.org/..Grade_K_-_8_Math_Curriculum_6-8-06....Numerical operations are an essential part of the mathematics curriculum, especially in .... Discrete Math Systematic Listing and Counting – Grade K Objectives: 1) To sort and ... Use hoola hoops or string to make Venn diagrams. Discrete Mat New Jersey Math Standards - 2nd Grade - MathScore.comhttps://www.mathscore.com/math/standards/.../2nd%20Grade/Whole numbers through hundreds (Counting Squares ) • Ordinals ... Use simple shapes to make designs, patterns, and pictures. 2. ... Part unknown (e.g., 3 + [] = 8) (Missing Term ) 4.3.2 D.4.4.2 C. Discrete Mathematics-Systematic Listing and Counting 1. Sort and classify objects according to attributes. • Venn diagrams 2.[DOC]Grade 6 andards - Arizona Department of Educationwww.azed.gov/standards-practices/files/2012/04/mathgrade6.docThey want to make a replica of the fish tank in their classroom but 4 times larger. .... is modeled by dividing the rectangle horizontally into 3 parts and then shading ..... Understand and demonstrate the systematic listing and counting of possible outcomes. .... The number 6 can be placed in the center of the Venn diagram as it ...
Counting:
In some questions you are asked to count something: How many apples did Jorge buy, how many taka did Shelly spend, how many pages were read, how many ways are there to complete a job, etc. Some such problems may be solved using algebra. Others may need your to systematically make a list. Sometimes you may need to know some special strategies that we discussed.
Making a systematic list:
When a question asks 'how many', often the best strategy is to make a list of all the possibilities. You have to make this list systematically so that no item is left out or so that you don't count an item twice. Usually this means listing the possibilities in a numerical or alphabetical order. Inmost case, shortly, After starting to make the systematic list, you world see a pattern developing and
you should be able to find out how many elements should be in the list without having to write drown all of them.
Example: A palindrome is a number, such a 93539, that reads the same forward and backward. How many palindromes are there between 100 and 1000?
Solution: At first let us write the palindromes between 100 and 200.
101, 111, 121,131,141,151, 161, 171, 181, 191 = 10 Palindromes
Next comes the number between 200 and 300.
202, 212, 222, 232, 242, 252, 262, 272, 282, 292 =10 Palindromes
So, we can see a pattern. Each interval of 100 numbers contains 10 Palindromes. So, beginning with 1, 2, 3,.. 9 there are 9*10 = 90 Palindromes.
Answer: 90
This a good technique if the numbers in the problem are small. Proper use of this technique eliminates the risk of marking an error in arithmetic.
Using Arithmetic to count:
Let us go through a few examples. These will illustrate the basic rules of arithmetic we use for counting..
Example A: Bari entered the store with $213 & left with $ 203. How much did he spend in the store?
Example B: A man sitting in a cinema hall ticket counter sold ticket no. 213 through 209. How many tickets were sold?
Example C: Bari is the 213th person in a queue and Shaki is the 209th person. How many people are there between them?
Solution: (A-c): At first sight it may seem that all the three have the same answer 213-209=4. But actually, the answer for A is 4. B is 5 and C is 3.
In example A, Bari had $ 213 He spent $ no. 213, 212, 211, 210. He still had $ 2.9 left with him. So, he spend $4.
In example B. tickets numbered 213, 212, 211, 210, & 209 were sold . So, the total number is 5.
In the queue (example C) between Bari & Shakily there are person no. 212, 211 & 210 . So, the answer is 3.
In example A, we just subtracted, in example B we subtracted & then added a 1, in example C we subtracted & then subtracted 1 more.
The main point to note is whether or not the first and last numbers are included. In example A, Bari left the store with the $ 209. In example B. both the 213 & 209 ticket were sold. And in example C. neither the 213 person nor the 209 person was counted. So, ultimately we come to the following rules:
A) If exactly one of the end points is included, subtract.
B) If both the end points are included, subtract and add1 .
C) If none of the end points are included, subtract & subtract 1 more.
PDF] Math - Douglas Unified School District www.dusd.k12.az.us/LinkClick.aspx?fileticket=uvt6LFtK2Gw%3D...counting. Students create and use multiple representations such as charts, systematic lists, and tree diagrams. They note ... meals can be made from the menu? Make a systematic list of your possibilities. How can you .... Use the Venn diagram to determine if the ... of square units by cutting the triangular parts from the. Discrete Mathematics - Systematic Listing and Counting.htm www.ntuaft.com/.../CPI%20Checklists/.../Discrete%20Mathematics%20-...C. Discrete Mathematics—Systematic Listing and Counting ... Venn diagrams. 2. ... Building upon knowledge and skills gained in preceding grades, by the end of ... Explore counting problems involving Venn diagrams with three attributes (e.g., ...NJ ASK Grade 7 Mathematics Test and Item Specifications www.state.nj.us/njded/assessment/ms/5-7/specs/math7.htm C. Discrete Mathematics—Systematic Listing and Counting ... We have to make sure that the raised expectations for all students do not result in ... Numerical operations are an essential part of the mathematics curriculum, especially in the ..... C.2 Explore counting problems involving Venn diagrams with three attributes (e.g., ... SticiGui Counting www.stat.berkeley.edu/~stark/SticiGui/Text/counting.htm... Sampling Distributions · Scatterplots · Student's t Distribution · Venn Diagram (2 subsets) ... This chapter presents techniques and shortcuts for counting that are ... to approach counting systematically: Enumerating all the possibilities becomes very ... Counting large sets by listing all the possibilities is ..... I'll make you rich!".Sets and Venn Diagrams - the Australian Mathematical Sciences ...www.amsi.org.au/teacher_modules/Sets_and_venn_diagrams.htmlSuch counting quickly throws up situations that may at first seem contradictory. ... The Venn diagram makes the situation easy to visualise. ... The five elements of the set are separated by commas, and the list is enclosed .... b: If the set S in each part is finite, write down | S |. ..... working systematically around the diagram. Counting Methods (with Videos) - Online Math Learning www.onlinemathlearning.com/counting-methods.html Sets and Lists, Venn Diagrams, Inclusion-Exclusion Principle, Multiplication ... Uncertainty is part of the process of making decisions and predicting outcomes. ... we now turn to a discussion of counting objects in a systematic way before ... [PDF] Probability and Counting Rules B&W CONFIRMINGS highered.mcgraw-hill.com/sites/dl/free/007353496x/366095/Ch_04.pdf time a person awakes until he or she goes to bed, that person makes decisions regarding the possible events that are governed at least in part by chance. ... explains how the counting rules and the probability rules can be used .... systematically. .... The Venn diagram that represents the probability of the complement of an ...[PDF] Matwww.bedminsterschool.org/..Grade_K_-_8_Math_Curriculum_6-8-06....Numerical operations are an essential part of the mathematics curriculum, especially in .... Discrete Math Systematic Listing and Counting – Grade K Objectives: 1) To sort and ... Use hoola hoops or string to make Venn diagrams. Discrete Mat New Jersey Math Standards - 2nd Grade - MathScore.comhttps://www.mathscore.com/math/standards/.../2nd%20Grade/Whole numbers through hundreds (Counting Squares ) • Ordinals ... Use simple shapes to make designs, patterns, and pictures. 2. ... Part unknown (e.g., 3 + [] = 8) (Missing Term ) 4.3.2 D.4.4.2 C. Discrete Mathematics-Systematic Listing and Counting 1. Sort and classify objects according to attributes. • Venn diagrams 2.[DOC]Grade 6 andards - Arizona Department of Educationwww.azed.gov/standards-practices/files/2012/04/mathgrade6.docThey want to make a replica of the fish tank in their classroom but 4 times larger. .... is modeled by dividing the rectangle horizontally into 3 parts and then shading ..... Understand and demonstrate the systematic listing and counting of possible outcomes. .... The number 6 can be placed in the center of the Venn diagram as it ...
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