Short Tricks to Solve Dice Sum Problem (2)

Duration: 12 min

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Inside: a video lesson and guided study material.

Module outline

  1. About Course: About the Course
  2. Section A: General Awareness: Polity, वैज्ञानिक उपलब्धियाँ और अनुसंधान / Scientific Achievements and Research, Biology, Chemistry, Static General Knowledge, History, Geography, Indian Economy, समसामयिकी / Current Affairs, Evolution of Indian Banking (Banking & Financial Awareness), Environment
  3. Section A: General Intelligence & Reasoning ability: श्रेणी / Series, कूट लेखन एवं कूट वाचन / Coding-Decoding, Data Sufficiency, अशाब्दिक तर्क / Non-Verbal Reasoning, Analytical Reasoning (Counting Figures Reasoning), रक्त संबंध / Blood Relations, Syllogisms, कथन आधारित तार्किक विवेचन / Statement-Based Reasoning, बैठक व्यवस्था एवं पहेलियाँ / Seating Arrangement and Puzzles, दिशा एवं दूरी / Direction and Distance, गणितीय संक्रियाएँ एवं लुप्त संख्या / Mathematical Operations and Missing Numbers, कैलेंडर एवं घड़ी / Calendar and Clock, Clock, घन, पासा एवं आकृतियों की गिनती / Cubes, Dice and Figure Counting, आँकड़ों की पर्याप्तता एवं निर्णय क्षमता / Data Sufficiency and Decision Making, वर्णमाला एवं संख्या परीक्षण / Alphabet and Number Test, Crypt Arithmetic, वेन आरेख, न्याय वाक्य एवं असमानता / Venn Diagrams, Syllogism and Inequalities, Data Arrangement, Inequality, सादृश्यता एवं वर्गीकरण / Analogy and Classification, क्रम एवं रैंकिंग / Order and Ranking, TCS NQT Advanced Reasoning Ability
  4. Section A: Arithmetical & Numerical Ability: अनुपात और समानुपात / Ratio and Proportion, Divisibility Rules, आँकड़ा निर्वचन / Data Interpretation, मिश्रण और एलिगेशन / Mixture and Alligation, Permutation and Combination, Logarithm, संख्या पद्धति / Number System, महत्तम समापवर्तक और लघुत्तम समापवर्त्य / HCF and LCM, Sequence and Series (Series), समय, चाल और दूरी / Time, Speed and Distance, प्रतिशत / Percentage, औसत / Average, लाभ, हानि और बट्टा / Profit, Loss and Discount, साधारण और चक्रवृद्धि ब्याज / Simple and Compound Interest, समय और कार्य / Time and Work, Pipe and Cistern, घन क्षेत्रमिति / Solid Mensuration, सरलीकरण, घातांक और करणी / Simplification, Indices and Surds, Algebra, Probability, Mental Ability, Age Problems, Partnership, Statistics, Arithmetic, Speed Mathematics and Basic Mathematics, Miscellaneous
  5. Section A: English Language & Comprehension: Vocabulary, Subject Verb Agreement (Verb Noun Agreement), Adjectives, Verbs and Tenses, Reading Comprehension, Parts of Speech, Sentence Construction, Prepositions and Conjunctions, Direct and Indirect Speech, Sentence Completion (Fill in the blanks), Sentence Structure and Common Errors, Modals, Transformation of Sentences, Sentence Correction (Error Correction), Adverb, Conjunction, Cloze Test (Cloze Passage)
  6. Section A: Hindi Language & Comprehension: भाषा, लिपि और वर्ण, वर्तनी, वर्ण-विन्यास, अनुस्वार, अनुनासिक, चंद्रबिंदु और ‘र’ के रूप, शब्द, पद और शब्द-भेद, संज्ञा और सर्वनाम, पद-भेद और रूप-विकार, लिंग, वचन, पुरुष और कारक, काल, वाच्य, वाक्य और वाक्य-भेद, संधि, समास, विलोम शब्द, पर्यायवाची शब्द, सामान्य अशुद्धियाँ, अनेक शब्दों के लिए एक शब्द, शब्द-भंडार, अपठित बोध, Multi-Topic Content (Cross-Concept Material), वर्णमाला
  7. Section B: Programming, Data Structures, OOP & Java: C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling, Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing, Java, OOPS with C++, Language Design, Intro to Compilers, Visual Basics, Syntax Analysis: Bottom-Up, Object Oriented Technology
  8. Section B: Computer Networks, TCP/IP & Network Security: Introduction to CN, Data Communication, DLL: Flow Control, DLL: Access Control, DLL: Error Control, Data Link Layer - Ethernet, Net Layer: IPv4 & Proto, Net Layer: IP Addressing, Transport Layer Services, TL: Congestion & UDP, Application Layer, Hardware basics, Network Access Technologies, Network Security, Society, Law & Ethics, Mobile Computing, Net Layer:Routing Protocol, Cyber Safety and Security
  9. Section B: Teaching Methodology: वृद्धि और विकास / Growth and Development, Concept of Pedagogy, विकास के सिद्धांत / Theories of Development, समावेशी शिक्षा / Inclusive Education, Experiential Pedagogy, आकलन और मूल्यांकन / Assessment and Evaluation, समाजीकरण और जेंडर / Socialisation and Gender, National Education Policy (NEP) 2020, संज्ञानात्मक प्रक्रियाएँ / Cognitive Processes, शिक्षण योजना और शिक्षण कौशल / Teaching Planning and Teaching Skills, Personality Development, शिक्षण विधियाँ / Teaching Methods, अधिगम और व्यवहारवादी सिद्धांत / Learning and Behavioural Theories, Learners & Learning Process, Factors Affecting Teaching, शिक्षण विधियाँ एवं उपागम / Teaching Methods and Approaches, शिक्षण-अधिगम प्रक्रिया एवं कक्षा प्रबंधन / Teaching-Learning Process and Classroom Management, शिक्षण सहायक सामग्री एवं शैक्षिक तकनीकी / Teaching Aids and Educational Technology, सूक्ष्म शिक्षण एवं कक्षा संप्रेषण / Microteaching and Classroom Communication, आकलन एवं मूल्यांकन / Assessment and Evaluation, पाठ्यचर्या, कला एवं सह-शैक्षिक क्षेत्र / Curriculum, Arts and Co-curricular Areas, दिव्यांग बच्चों की शिक्षा / Education of Children with Disabilities, शिक्षक: गुण, व्यावसायिक नैतिकता एवं विकास / Teacher: Qualities, Professional Ethics and Development
  10. Section B: Digital Electronics / Boolean Logic: Digital Systems & Boolean Basics, Logic Gates & Hardware, Boolean Expression, Boolean Minimization, Combinational Circuit, Sequential Circuits, Number System, Number Representation, Analog-Digital Conversion
  11. Section B: Operating Systems & Linux: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, File Management, Disc Scheduling, Unix OS, Linux OS, Distributed Systems, Windows OS
  12. Section B: Data Base Management System: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, Relational Algebra, SQL, File Organization & Indexing, Transaction Management, Concurrency Control, Database Security & Authorization, Database Recovery, Data Warehousing & Mining
  13. Section B: Internet & Web Technologies: Website, Browser & Services, Internet Fundamentals, Web Design & Development, Protocols & Services, Web Publishing & Hosting, Server Side & Security, HTML Basics, CSS, Emerging Technologies, JavaScript-Basics, .Net Programming, Electronic Data Interchange and E-Commerce, XML
  14. Section B: Computer Fundamentals, IT Basics & Architecture: Basics of COA, Cache Memory Organization, Input Output Organisation, Register Transfer and Microoperations, Control Unit Design, Programming the Basic Computer, Terms & Abbreviations, MS Office Applications, Instr Formats & Modes, Floating Point Rep, कंप्यूटर के मूल सिद्धांत / Computer Fundamentals, साइबर सुरक्षा / Cyber Safety and Security, कंप्यूटर सॉफ़्टवेयर / Computer Software, ऑपरेटिंग सिस्टम / Operating System, वर्ड प्रोसेसिंग (MS Word) / Word Processing (MS Word), कंप्यूटर नेटवर्क / Computer Networks, प्रेज़ेंटेशन (MS PowerPoint) / Presentation Software (MS PowerPoint)
  15. Section B: Python Language: Introduction to Python, Tokens, Variables, Data Types, I/O, Operators & Expressions, Control Flow & Looping, Collections & Strings, Functions & Modules, File Handling & Data Analysis
  16. Section B: Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Testing, Software Maintenance and Configuration Management, extras, Fundamentals of Systems and Information Systems
  17. Section B: Design & Analysis of Algorithms: Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos, Advanced Algorithms
  18. Section B: Mathematics: Linear Algebra, Calculus, Trigonometry, Vectors, Set Theory, Functions, Relations, Propositional and Predicate Logic, Research Process Steps, Graph Theory, समुच्चय, संबंध, फलन और विविक्त गणित / Sets, Relations, Functions and Discrete Mathematics, संख्या पद्धति और संख्या सिद्धांत / Number Systems and Number Theory, यूक्लिडीय ज्यामिति और क्षेत्रमिति / Euclidean Geometry and Mensuration, सांख्यिकी और प्रायिकता / Statistics and Probability, निर्देशांक ज्यामिति, सदिश और त्रिविमीय ज्यामिति / Coordinate Geometry, Vectors and 3D Geometry, बहुपद, समीकरण और सम्मिश्र संख्याएँ / Polynomials, Equations and Complex Numbers, समतल क्षेत्रमिति / Plane Mensuration, प्रतिचयन / Sampling
  19. Section B: Computer Graphics, Multimedia & Mobile Computing: Basics of Computer Graphics, 2-D Geometrical Transforms and Viewing, 3-D Object Representation, Geometric Transformations and Viewing, Multimedia
  20. Section B: Emerging Trends: Emerging Tech Trends, Knowledge Representation, Natural Language Processing, व्यापार, विपणन एवं उपभोक्ता संरक्षण / Trade, Marketing and Consumer Protection
  21. DSSSB Previous Year Paper:
  22. Business Studies and Management: Business Communication, प्रबंध के सिद्धांत एवं कार्य / Principles and Functions of Management, लेखांकन के आधार एवं लेन-देनों का अभिलेखन / Accounting Fundamentals and Recording of Transactions, Business Economics, ह्रास, अंतिम खाते व कंप्यूटरीकृत लेखांकन / Depreciation, Final Accounts and Computerised Accounting, Knowledge Management & New Economy
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This educational video is a mathematics lecture that explains how to solve a classic combinatorics problem: finding the number of ways to get a sum of 10 when three dice are thrown. The video begins with a title slide that introduces the topic of permutations and combinations, setting the context for the problem. The instructor then presents the problem statement: 'If three dice are thrown, find the number of ways to get sum 10?'. The core of the lesson involves a methodical approach using the stars and bars technique, but with constraints. The variables x1, x2, and x3 are defined as the values on the three dice, with the equation x1 + x2 + x3 = 10. The constraint 1 ≤ xi ≤ 6 is applied to each variable, as a die cannot show a value less than 1 or more than 6. The instructor demonstrates the standard method for solving such problems by first transforming the variables to remove the lower bound (by setting xi' = xi - 1), which leads to the equation x1' + x2' + x3' = 7. The total number of non-negative integer solutions is calculated as C(7+3-1, 3-1) = C(9,2). To account for the upper bound constraint, the instructor uses the principle of inclusion and exclusion. The method involves subtracting the cases where one or more variables exceed 6. For example, if x1 > 6, a substitution x1'' = x1 - 6 is made, leading to a new equation x1'' + x2' + x3' = 1. The number of solutions for this case is C(1+3-1, 3-1) = C(3,2). Since there are three dice, this case is multiplied by 3. The final answer is the total number of solutions minus the invalid cases: C(9,2) - 3*C(3,2) = 36 - 9 = 27. The video concludes with a general formula for the number of ways to get a sum n with three dice, which is C(n-1,2) - 3*C(n-7,2) for n from 7 to 12, and a final 'Thanks for watching' screen.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video opens with a title slide that introduces the topic of permutations and combinations. The slide is divided into four quadrants, each with a visual metaphor: a green craps table with dice for 'PERMUTATION', a basket of fruit for 'COMBINATION', a pizza for 'COMBINATION', and two men in a business meeting for 'PERMUTATION'. This is followed by a presentation slide titled 'Confused' which poses the question, 'Should I unlock with Permutation or Combination?'. The slide features a cartoon detective with a magnifying glass, standing between two locked doors labeled 'Permutation' and 'Combination', visually representing the confusion between the two concepts. The instructor, Yash Jain, is visible in a small window in the bottom right corner. The slide also includes the 'Knowledge Gate Educator' logo and branding.

  2. 2:00 – 5:00 02:00-05:00

    The video transitions to a new slide titled 'Dice Sum Problem'. The instructor presents the core problem: 'If three dice are thrown, find the number of ways to get sum 10?'. The instructor begins to set up the mathematical model by defining the variables x1, x2, and x3 as the values on the three dice. The equation x1 + x2 + x3 = 10 is written on the screen. The constraints 1 ≤ x1 ≤ 6, 1 ≤ x2 ≤ 6, and 1 ≤ x3 ≤ 6 are also written, reflecting the possible values on a standard die. The instructor explains that this is a problem of finding the number of integer solutions to this equation under the given constraints.

  3. 5:00 – 10:00 05:00-10:00

    The instructor explains the method to solve the problem. He first removes the lower bound constraint by substituting xi' = xi - 1, which transforms the equation to x1' + x2' + x3' = 7, where xi' ≥ 0. He then uses the stars and bars method to find the total number of non-negative integer solutions, which is C(7+3-1, 3-1) = C(9,2). He then addresses the upper bound constraint. He explains that the total count includes invalid cases where one of the variables is greater than 6. He demonstrates the inclusion-exclusion principle by calculating the number of invalid cases. For example, if x1 > 6, he substitutes x1'' = x1 - 6, leading to the equation x1'' + x2' + x3' = 1. The number of solutions for this is C(1+3-1, 3-1) = C(3,2). Since there are three dice, this case is multiplied by 3. The final answer is the total number of solutions minus the invalid cases: C(9,2) - 3*C(3,2) = 36 - 9 = 27.

  4. 10:00 – 11:47 10:00-11:47

    The instructor generalizes the solution. He writes the formula for the number of ways to get a sum n with three dice as C(n-1,2) - 3*C(n-7,2) for n from 7 to 12. He explains that this formula accounts for the total number of solutions and subtracts the cases where one of the dice shows a value greater than 6. The video concludes with a black screen displaying the text 'THANKS FOR WATCHING' in an orange and white box.

The video provides a clear, step-by-step tutorial on solving a combinatorics problem involving dice. It begins by establishing the context of permutations and combinations, then focuses on a specific problem. The core of the lesson is the application of the stars and bars method to find the number of integer solutions to an equation, with a crucial emphasis on handling constraints. The instructor demonstrates the principle of inclusion and exclusion to subtract invalid cases where a die value exceeds 6. The progression from setting up the problem to applying the formula and generalizing the result provides a comprehensive understanding of the method, making it a valuable resource for students learning combinatorics.

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