Short Tricks to Solve Dice Sum Problem (2)

Duration: 12 min

This video lesson is available to enrolled students.

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  3. Paper 1 | General Hindi: भाषा, लिपि, ध्वनि, वर्ण, अक्षर, स्वर, व्यंजन, मात्रा और संयुक्ताक्षर, वर्तनी, वर्ण-विन्यास, अनुस्वार, अनुनासिक, चंद्रबिंदु और ‘र’ के रूप, शब्द, पद और शब्द-भेद, हिन्दी की रूप रचना, संज्ञा और सर्वनाम, विशेषण और क्रिया, लिंग, वचन, पुरुष और कारक, काल, वाच्य, वाक्य और वाक्य-भेद, वर्णमाला, समास, संधि, Multi-Topic Content (Cross-Concept Material), हिन्दी की ध्वनि और स्वनिम व्यवस्था, प्राचीन और मध्यकालीन भारतीय आर्यभाषाएँ, काव्य के लक्षण, हेतु और प्रयोजन, हिन्दी शब्द रचना, हिन्दी साहित्येतिहास दर्शन, भक्तिकालीन काव्यधाराएँ, भारतीय काव्यशास्त्र की परंपरा, भारतीय नवजागरण और स्वाधीनता आंदोलन, अन्य गद्य विधाएँ एवं प्रवासी साहित्य, निर्धारित उपन्यासों का तुलनात्मक अध्ययन, गोदान, हिन्दी कहानी का विकास, स्वातंत्र्योत्तर और नयी कहानी, हिन्दी निबंध का विकास, शुक्लोत्तर निबंध, द्विवेदी और शुक्ल युग के निबंध, हिन्दी की संवैधानिक स्थिति, आधुनिक भारतीय आर्यभाषाएँ और हिन्दी का उद्भव, हिन्दी भाषा प्रयोग के विविध रूप, रीतिकाल, जीवनी, द्विवेदी युग और छायावाद, रीति, वक्रोक्ति और औचित्य सिद्धान्त, शब्दशक्ति और ध्वनि सिद्धान्त, रस सिद्धान्त और साधारणीकरण, उत्तर आधुनिकतावाद, लोहिया दर्शन, हिन्दी उपन्यास का विकास, हिन्दी नवजागरण, अंबेडकर दर्शन, आदिकाल, प्रयोगवाद और नयी कविता, छायावादी काव्य: प्रसाद और निराला, प्रगतिवाद से समकालीन कविता, निर्गुण भक्ति काव्य, द्विवेदीयुगीन काव्य, रीतिकालीन काव्य, आरंभिक हिन्दी उपन्यास, स्वातंत्र्योत्तर उपन्यास, आत्मकथा, स्वाधीनता-पूर्व प्रेमचंदोत्तर उपन्यास, निर्धारित कहानियों का तुलनात्मक अध्ययन, साठोत्तरी नाटक, प्रसाद और प्रसादोत्तर युग के नाटक, भारतेन्दु युगीन नाटक, स्वातंत्र्योत्तर नाटक, भारतेन्दु युगीन निबंध, संस्मरण और रेखाचित्र, व्यंग्य, एक साहित्यिक की डायरी, संस्कृति के चार अध्याय, सगुण भक्ति काव्य, हिन्दी नाटक, एकांकी और रंगमंच, आधुनिक गद्य और हिन्दी नवजागरण, अलंकार सिद्धान्त, हिन्दी आलोचना का विकास, भक्ति आंदोलन, कल्पना, बिम्ब, प्रतीक, मिथक और फंतासी, नयी कविता का विस्तार और साठोत्तरी कविता, उत्तर-छायावादी और प्रगतिवादी काव्य, काव्यगुण और काव्यदोष, प्लेटो, अरस्तू और पाश्चात्य परंपरा, मार्क्सवाद, स्वच्छन्दतावादी काव्यशास्त्र, आधुनिक पाश्चात्य आलोचना, भाषा और भाषा परिवार, देवनागरी लिपि
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  5. Paper 1 | Programming Language: Language Design, C Fundamentals, Control Flow, Functions, Arrays & Pointers, Storage Classes, Structures & Enums, DMA, Macros, Scoping & File Handling, OOPS with C++
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  7. Paper 1 | Cyber Security: कंप्यूटर की मूल बातें और हार्डवेयर / Computer Fundamentals and Hardware, प्रत्यक्ष बोध / Literal Comprehension, कंप्यूटर सॉफ्टवेयर / Computer Software, निष्कर्षात्मक बोध / Inferential Comprehension, आलोचनात्मक बोध / Critical Comprehension, इंटरनेट और वर्ल्ड वाइड वेब / Internet and World Wide Web, ई-मेल और ऑनलाइन संचार / Email and Online Communication, साइबर सुरक्षा / Cyber Security, अपशिष्ट प्रबंधन / Waste Management, डिजिटल नैतिकता, बौद्धिक संपदा और समाज / Digital Ethics, IPR and Society, डिजिटल एवं नव माध्यम / Digital and New Media, शोध नैतिकता / Research Ethics, उभरती प्रौद्योगिकियाँ / Emerging Technologies, शिक्षा और शासन में ICT / ICT in Education and Governance, Society, Law & Ethics, Cyber Threats and Malware Attacks, Role of Internet and Web Services, Security in Mobile and Wireless Computing, Electronic Data Interchange and E-Commerce, Physical Security and Cryptography Basics, Intrusion Detection Systems (IDS), Cyber Safety and Security, Ethical Hacking, भौगोलिक संकेतों का संरक्षण / Protection of Geographical Indications
  8. Paper 1 | Computer Organization and Architecture: Basics of COA, Programming the Basic Computer, Cache Memory Organization, Input Output Organisation, Instr Formats & Modes, Register Transfer and Microoperations, Control Unit Design
  9. Paper 1 | Computer Networks: Introduction to CN, Data Communication, DLL: Access Control, DLL: Flow Control, DLL: Error Control, DLL: Framing, Data Link Layer - Ethernet, Net Layer: IPv4 & Proto, Net Layer: IP Addressing, Net Layer:Routing Protocol, Transport Layer Services, TL: Congestion & UDP, Application Layer, Hardware basics, Network Security
  10. Paper 1 | Design and Analysis of Algorithms: Algorithm Analysis, Time Complexity Analysis, Sorting Algorithms, Greedy Algorithms, Dynamic Programming, Minimum Spanning Trees, Shortest Path Algos, Advanced Algorithms
  11. Paper 1 | Data Structures: Introduction to DS, Array, Linked List, Stack, Queue, Tree, Graphs, Hashing
  12. Paper 1 | Discrete Structures and Theory of Logic: क्रमचय, संचय एवं प्रायिकता / Permutation, Combination and Probability, Set Theory, Relations, Functions, Propositional and Predicate Logic, Graph Theory, Group Theory
  13. Paper 1 | Software Project Management: Software Project Management
  14. Paper 2 | Operating Systems: Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Threads & Process Creation, Deadlock, Memory Management, Virtual Memory, Disc Scheduling, File Management, Unix OS, Linux OS, Security
  15. Paper 2 | Compiler Design: Intro to Compilers, Lexical Analysis, Grammar & CFG, Syntax Analysis: Top-Down, Syntax Analysis: Bottom-Up, Semantic Analysis & SDT, Intermediate Code Gen, Code Optimization, Run Time Environment, Introduction to TOC, Deterministic FA (DFA), Non-Deterministic FA, Regular Expressions, Pushdown Automata & CFG, Complexity Theory
  16. Paper 2 | Artificial Intelligence & High Performance Computing: Approaches to AI, Search Algorithms, Game Playing, Knowledge Representation, Planning, Multi Agent Systems, Natural Language Processing, क्लाउड और वितरित कंप्यूटिंग / Cloud and Distributed Computing, Emerging Tech Trends
  17. Paper 2 | Image Processing: Digital Image Fundamentals, Image Enhancement, Image Restoration, Color Image Processing, Image Compression
  18. Paper 2 | Software Engineering: Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Design, Estimation and Metrics, Software Testing, Software Maintenance and Configuration Management
  19. Paper 2 | Computer Graphics: Basics of Computer Graphics, 2-D Geometrical Transforms and Viewing, 3-D Object Representation, Geometric Transformations and Viewing
  20. Paper 2 | Web Technologies: Website, Browser & Services, Internet Fundamentals, Web Design & Development, Protocols & Services, Web Publishing & Hosting, Server Side & Security, Emerging Technologies, HTML Basics, JavaScript-Basics, Java
  21. Paper 2 | Soft Computing: Fuzzy Sets, Artificial Neural Networks, Genetic Algorithms
  22. Paper 2 | Distributed System: Distributed Systems
  23. UPPSC Live Class Recordings: UPPSC Polytechnic Lecturer 2025 (CS)
  24. Previous Year Papers:
  25. Mock Test:
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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