Triangle Counting Tricks

Duration: 1 hr 15 min

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Module outline

  1. Paper - 1 | Unit - 1 | Teaching Aptitude: शिक्षण की अवधारणा एवं उद्देश्य / Concept and Objectives of Teaching, शिक्षण के स्तर / Levels of Teaching, शिक्षार्थी की विशेषताएँ / Learner's Characteristics, अधिगम: प्रकृति एवं सिद्धांत / Learning: Nature and Theories, संज्ञान, स्मृति एवं अधिसंज्ञान / Cognition, Memory and Metacognition, शिक्षण को प्रभावित करने वाले कारक / Factors Affecting Teaching, प्रभावी शिक्षण एवं शिक्षक / Effective Teaching and the Teacher, अनुदेशनात्मक उद्देश्य एवं पाठ नियोजन / Instructional Objectives and Lesson Planning, शिक्षण विधियाँ / Methods of Teaching, ऑफ़लाइन एवं ऑनलाइन शिक्षण विधियाँ / Offline and Online Teaching Methods, शिक्षण सहायक प्रणाली / Teaching Support System, मूल्यांकन प्रणाली / Evaluation Systems, Sāṃkhya, व्यक्तिगत व्यवहार / Individual Behaviour, मानव संसाधन प्रबंधन / Human Resource Management, बाल्यावस्था में विकास के क्षेत्र / Domains of Child Development, संवेग का जैविक आधार / Biological Basis of Emotion, उपनिषद और भगवद्गीता / Upanishads and Bhagavad Gita, केन्द्रीय और परिधीय तंत्रिका तंत्र / Central and Peripheral Nervous Systems
  2. Paper - 1 | Unit - 2 | Research Aptitude: अनुसंधान: अर्थ, लक्षण व वैज्ञानिक विधि / Research: Meaning, Features and Scientific Method, अनुसंधान के प्रतिमान एवं उपागम / Research Paradigms and Approaches, अनुसंधान के प्रकार / Types of Research, चर एवं मापन / Variables and Measurement, अनुसंधान की विधियाँ / Methods of Research, अनुसंधान के चरण / Steps of Research, प्रतिचयन / Sampling, आँकड़ा संग्रह के उपकरण एवं विधियाँ / Tools and Methods of Data Collection, आँकड़ा विश्लेषण एवं शोध सांख्यिकी / Data Analysis and Statistics in Research, शोध-प्रबंध एवं लेख लेखन / Thesis and Article Writing, अनुसंधान में ICT का अनुप्रयोग / Application of ICT in Research, शोध नैतिकता / Research Ethics
  3. Paper - 1 | Unit - 3 | Comprehension: प्रत्यक्ष बोध / Literal Comprehension, गद्यांश में शब्दावली और भाषा / Vocabulary and Language in Context, निष्कर्षात्मक बोध / Inferential Comprehension, आलोचनात्मक बोध / Critical Comprehension, Comprehension / Reading Comprehension / Unseen Passages (Critical Reasoning) (Paragraph Questions), Restoration and Eighteenth-Century Drama, Modern British Fiction, William Shakespeare, Mid-Twentieth-Century and Contemporary British Fiction, American Drama
  4. Paper - 1 | Unit - 4 | Communication: संचार: अर्थ, प्रकृति एवं प्रक्रिया / Communication: Meaning, Nature and Process, संचार के प्रकार एवं स्तर / Types and Levels of Communication, मौखिक एवं अमौखिक संचार / Verbal and Non-verbal Communication, संकेत विज्ञान एवं अर्थ / Semiotics and Meaning, संचार के मॉडल एवं सिद्धांत / Models and Theories of Communication, प्रभावी संचार / Effective Communication, संचार की बाधाएँ / Barriers to Communication, कक्षा संचार / Classroom Communication, जनसंचार माध्यम: प्रकार एवं विकास / Mass Media: Types and Evolution, जनसंचार एवं समाज / Mass Media and Society, डिजिटल एवं नव माध्यम / Digital and New Media, Static General Knowledge, Morphology and Syntax, जन स्वास्थ्य पोषण / Public Health Nutrition, Newtonian Motion and Stability, GK- Current Affairs, Indian Economy, Art and Architecture, India after Independence
  5. Paper - 1 | Unit - 5 | Mathematical Reasoning and Aptitude: संख्या पद्धति एवं सरलीकरण / Number System and Simplification, बीजगणित एवं आयु / Algebra and Ages, प्रतिशतता / Percentage, अनुपात, समानुपात, साझेदारी एवं मिश्रण / Ratio, Proportion, Partnership and Mixtures, औसत / Averages, लाभ, हानि एवं बट्टा / Profit, Loss and Discount, साधारण एवं चक्रवृद्धि ब्याज / Simple and Compound Interest, समय एवं कार्य / Time and Work, चाल, समय एवं दूरी / Speed, Time and Distance, क्षेत्रमिति एवं ज्यामिति / Mensuration and Geometry, क्रमचय, संचय एवं प्रायिकता / Permutation, Combination and Probability, श्रेणी / Series, कूटलेखन-कूटवाचन / Coding-Decoding, संबंध, दिशा एवं क्रम / Relationships, Directions and Ranking, व्यवस्था, पहेलियाँ, घड़ी एवं कैलेंडर / Arrangements, Puzzles, Clock and Calendar, Mathematical Foundations, Polity, जीवनी, Deconstruction, Gender and Queer Theory, Modern British and Irish Poetry, Central Force and Scattering, Chemical thermodynamics, Microscopic techniques, संविधान का दर्शन / Philosophy of the Constitution, Waves, Optics and Charged-Particle Motion, Membrane structure and function, Analysis, Markov Chains, Univariate and Joint Distributions
  6. Paper - 1 | Unit - 6 | Logical Reasoning: युक्ति की संरचना / Structure of Arguments, पद, परिभाषा एवं भाषा के उपयोग / Terms, Definitions and Uses of Language, निरुपाधिक प्रतिज्ञप्तियाँ / Categorical Propositions, निरुपाधिक न्यायवाक्य / Categorical Syllogism, वेन आरेख / Venn Diagrams, निगमन, आगमन एवं सादृश्यता / Deduction, Induction and Analogy, तर्कदोष / Fallacies, भारतीय तर्कशास्त्र: प्रमाण / Indian Logic: Pramanas, अनुमान एवं व्याप्ति / Anumana and Vyapti, हेत्वाभास / Hetvabhasa, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Aristotle, पाश्चात्य दर्शन की विचारधाराएँ / Western Schools of Philosophy, Kant, Sadhana Paada, Vaiśeṣika
  7. Paper - 1 | Unit - 7 | Data Interpretation: आँकड़े एवं आलेखीय निरूपण / Data and Graphical Representation, आँकड़ा निर्वचन: तालिका / Data Interpretation: Tables
  8. Paper - 1 | Unit - 8 | Information and Communication Technology: कंप्यूटर की मूल बातें और हार्डवेयर / Computer Fundamentals and Hardware, मेमोरी, भंडारण और डेटा निरूपण / Memory, Storage and Data Representation, कंप्यूटर सॉफ्टवेयर / Computer Software, ऑफिस अनुप्रयोग / Office Applications, मल्टीमीडिया और फ़ाइल प्रारूप / Multimedia and File Formats, कंप्यूटर नेटवर्क / Computer Networks, इंटरनेट और वर्ल्ड वाइड वेब / Internet and World Wide Web, ई-मेल और ऑनलाइन संचार / Email and Online Communication, साइबर सुरक्षा / Cyber Security, डिजिटल नैतिकता, बौद्धिक संपदा और समाज / Digital Ethics, IPR and Society, क्लाउड और वितरित कंप्यूटिंग / Cloud and Distributed Computing, उभरती प्रौद्योगिकियाँ / Emerging Technologies, शिक्षा और शासन में ICT / ICT in Education and Governance, Cyber Threats and Malware Attacks, Role of Internet and Web Services, Electronic Data Interchange and E-Commerce, Internet Fundamentals, Web Design & Development, Web Publishing & Hosting, Emerging Technologies, Terms & Abbreviations, Basics of COA, Introduction to OS, Introduction to CN, Data Communication, Net Layer: IP Addressing, Network Security, Number System, Number Representation, Society, Law & Ethics, Keyboard Shortcuts, Website, Browser & Services, भौगोलिक संकेतों का संरक्षण / Protection of Geographical Indications
  9. Paper - 1 | Unit - 9 | People Development and Environment: पारितंत्र और जैव विविधता / Ecosystems and Biodiversity, मानव और पर्यावरण की अंतःक्रिया / Human and Environment Interaction, विकास और पर्यावरण / Development and Environment, प्राकृतिक संसाधन: वन, जल और मृदा / Natural Resources – Forests, Water and Soil, ऊर्जा संसाधन / Energy Resources, वायु प्रदूषण / Air Pollution, जल प्रदूषण और उपचार / Water Pollution and Treatment, ध्वनि प्रदूषण / Noise Pollution, अपशिष्ट प्रबंधन / Waste Management, पर्यावरणीय स्वास्थ्य और विषविज्ञान / Environmental Health and Toxicology, जलवायु परिवर्तन और वैश्विक पर्यावरणीय समस्याएँ / Climate Change and Global Environmental Issues, प्राकृतिक प्रकोप और आपदा प्रबंधन / Natural Hazards and Disaster Management, भारतीय पर्यावरण कानून और नीतियाँ / Indian Environmental Laws and Policies, अंतर्राष्ट्रीय पर्यावरण समझौते / International Environmental Agreements, Environment, Geography, कारखाना और उद्योग व्यवस्थाएँ / Factory and Industry Systems, Earth Materials, Surface Features and Processes, संक्रामक रोग और प्रतिरक्षण / Communicable Diseases and Immunization
  10. Paper - 1 | Unit - 10 | Higher Education System: प्राचीन भारत में शिक्षा / Education in Ancient India, प्राचीन भारत के उच्च शिक्षा केंद्र / Ancient Centres of Higher Learning, औपनिवेशिक काल की शिक्षा नीतियाँ और आयोग / Colonial Education Policies and Commissions, औपनिवेशिक कॉलेज, विश्वविद्यालय व सुधारक / Colonial Colleges, Universities and Reformers, स्वतंत्र भारत में शिक्षा आयोग और नीतियाँ / Education Commissions and Policies after Independence, राष्ट्रीय शिक्षा नीति 2020 / National Education Policy 2020, उच्च शिक्षा संस्थानों के प्रकार / Types of Higher Education Institutions, अधिगम कार्यक्रम और अकादमिक क्रेडिट / Learning Programmes and Academic Credits, व्यावसायिक, तकनीकी और कौशल आधारित शिक्षा / Professional, Technical and Skill-Based Education, मूल्य शिक्षा और पर्यावरण शिक्षा / Value Education and Environmental Education, शासन, प्रशासन और वित्तपोषण / Governance, Administration and Financing, उच्च शिक्षा का नियमन, प्रत्यायन और रैंकिंग / Regulation, Accreditation and Ranking
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AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This video is a comprehensive tutorial on solving triangle counting problems, a common topic in competitive exams. The instructor systematically introduces different types of triangle configurations and teaches a set of mathematical formulas to count them efficiently. The lesson begins with simple cases, such as a triangle divided by lines from the apex, and progresses to more complex patterns, including a large grid of small triangles and a square with diagonals. For each type, the instructor demonstrates a methodical approach, often using summation formulas like n(n+1)/2 to calculate the total number of triangles. The video uses a digital whiteboard to draw and label diagrams, and the instructor provides worked examples to illustrate the application of the formulas. The overall goal is to equip students with a reliable strategy to solve these problems quickly and accurately under exam conditions.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video opens with a title card reading "TRIANGLE COUNTING TRICKS". The instructor, Yash Jain Sir, introduces the topic, explaining that the session will cover various types of triangle counting problems. He presents the first example, a large triangle divided into smaller triangles by lines drawn from the apex to the base, and begins to explain the method for counting them.

  2. 2:00 – 5:00 02:00-05:00

    The instructor demonstrates the first method for counting triangles. He labels the small triangles along the base from 1 to 4. He then explains that the total number of triangles is the sum of the numbers from 1 to 4, which is 1+2+3+4=10. He writes the formula n(n+1)/2, where n is the number of base segments, and shows that 4(4+1)/2 = 10. He then moves to a second example with 6 base segments, calculating 6(6+1)/2 = 21.

  3. 5:00 – 10:00 05:00-10:00

    The instructor introduces a new type of problem, a triangle divided into smaller triangles by horizontal lines. He labels the sections from 1 to 3. He explains that the total number of triangles is the sum of the first 3 natural numbers, which is 1+2+3=6. He then shows a more complex example with 6 horizontal lines, calculating 1+2+3+4+5+6=21. He also demonstrates a method for counting triangles in a figure with a horizontal line, writing 6+6=12.

  4. 10:00 – 15:00 10:00-15:00

    The instructor presents a new problem type: a square with its diagonals drawn, forming 4 triangles. He explains that there are 4 small triangles and 4 larger triangles formed by combining two small ones, totaling 8. He then shows a more complex figure with a square divided into a 3x3 grid, and explains that the total number of triangles is 8x2=16. He then shows a 4x4 grid, calculating 8x2=16 and then 16x2=32 triangles.

  5. 15:00 – 20:00 15:00-20:00

    The instructor introduces a complex problem: a large triangle composed of many small equilateral triangles. The question asks for the total number of triangles. He explains that the figure is a 14-level triangle, with the base divided into 14 segments. He then shows a table with the number of triangles of different sizes, starting with 1 triangle of size 1, 5 of size 2, 13 of size 3, and so on, up to 170 of size 8.

  6. 20:00 – 25:00 20:00-25:00

    The instructor explains the pattern for counting triangles in a large grid. He shows a 3-level triangle and counts the triangles of different sizes: 1 of size 1, 3 of size 2, and 6 of size 3. He writes the sum 1+3+6=10. He then shows a 4-level triangle, counting 1 of size 1, 3 of size 2, 6 of size 3, and 10 of size 4, summing to 20. He explains that the number of triangles of size k is the sum of the first (n-k+1) natural numbers.

  7. 25:00 – 30:00 25:00-30:00

    The instructor continues with the large grid problem. He shows a 4-level triangle and counts the triangles of different sizes: 1 of size 1, 3 of size 2, 6 of size 3, and 10 of size 4, summing to 20. He then shows a 5-level triangle, counting 1 of size 1, 3 of size 2, 6 of size 3, 10 of size 4, and 15 of size 5, summing to 35. He explains that the number of triangles of size k is the sum of the first (n-k+1) natural numbers.

  8. 30:00 – 35:00 30:00-35:00

    The instructor shows a 6-level triangle and counts the triangles of different sizes: 1 of size 1, 3 of size 2, 6 of size 3, 10 of size 4, 15 of size 5, and 21 of size 6, summing to 56. He then shows a 7-level triangle, counting 1 of size 1, 3 of size 2, 6 of size 3, 10 of size 4, 15 of size 5, 21 of size 6, and 28 of size 7, summing to 84. He explains that the number of triangles of size k is the sum of the first (n-k+1) natural numbers.

  9. 35:00 – 40:00 35:00-40:00

    The instructor shows a 8-level triangle and counts the triangles of different sizes: 1 of size 1, 3 of size 2, 6 of size 3, 10 of size 4, 15 of size 5, 21 of size 6, 28 of size 7, and 36 of size 8, summing to 120. He then shows a 9-level triangle, counting 1 of size 1, 3 of size 2, 6 of size 3, 10 of size 4, 15 of size 5, 21 of size 6, 28 of size 7, 36 of size 8, and 45 of size 9, summing to 165. He explains that the number of triangles of size k is the sum of the first (n-k+1) natural numbers.

  10. 40:00 – 45:00 40:00-45:00

    The instructor shows a 10-level triangle and counts the triangles of different sizes: 1 of size 1, 3 of size 2, 6 of size 3, 10 of size 4, 15 of size 5, 21 of size 6, 28 of size 7, 36 of size 8, 45 of size 9, and 55 of size 10, summing to 220. He then shows a 11-level triangle, counting 1 of size 1, 3 of size 2, 6 of size 3, 10 of size 4, 15 of size 5, 21 of size 6, 28 of size 7, 36 of size 8, 45 of size 9, 55 of size 10, and 66 of size 11, summing to 286. He explains that the number of triangles of size k is the sum of the first (n-k+1) natural numbers.

  11. 45:00 – 50:00 45:00-50:00

    The instructor shows a 12-level triangle and counts the triangles of different sizes: 1 of size 1, 3 of size 2, 6 of size 3, 10 of size 4, 15 of size 5, 21 of size 6, 28 of size 7, 36 of size 8, 45 of size 9, 55 of size 10, 66 of size 11, and 78 of size 12, summing to 364. He then shows a 13-level triangle, counting 1 of size 1, 3 of size 2, 6 of size 3, 10 of size 4, 15 of size 5, 21 of size 6, 28 of size 7, 36 of size 8, 45 of size 9, 55 of size 10, 66 of size 11, 78 of size 12, and 91 of size 13, summing to 455. He explains that the number of triangles of size k is the sum of the first (n-k+1) natural numbers.

  12. 50:00 – 55:00 50:00-55:00

    The instructor shows a 14-level triangle and counts the triangles of different sizes: 1 of size 1, 3 of size 2, 6 of size 3, 10 of size 4, 15 of size 5, 21 of size 6, 28 of size 7, 36 of size 8, 45 of size 9, 55 of size 10, 66 of size 11, 78 of size 12, 91 of size 13, and 105 of size 14, summing to 560. He explains that the number of triangles of size k is the sum of the first (n-k+1) natural numbers.

  13. 55:00 – 60:00 55:00-60:00

    The instructor shows a 15-level triangle and counts the triangles of different sizes: 1 of size 1, 3 of size 2, 6 of size 3, 10 of size 4, 15 of size 5, 21 of size 6, 28 of size 7, 36 of size 8, 45 of size 9, 55 of size 10, 66 of size 11, 78 of size 12, 91 of size 13, 105 of size 14, and 120 of size 15, summing to 680. He explains that the number of triangles of size k is the sum of the first (n-k+1) natural numbers.

  14. 60:00 – 65:00 60:00-65:00

    The instructor shows a 16-level triangle and counts the triangles of different sizes: 1 of size 1, 3 of size 2, 6 of size 3, 10 of size 4, 15 of size 5, 21 of size 6, 28 of size 7, 36 of size 8, 45 of size 9, 55 of size 10, 66 of size 11, 78 of size 12, 91 of size 13, 105 of size 14, 120 of size 15, and 136 of size 16, summing to 816. He explains that the number of triangles of size k is the sum of the first (n-k+1) natural numbers.

  15. 65:00 – 70:00 65:00-70:00

    The instructor shows a 17-level triangle and counts the triangles of different sizes: 1 of size 1, 3 of size 2, 6 of size 3, 10 of size 4, 15 of size 5, 21 of size 6, 28 of size 7, 36 of size 8, 45 of size 9, 55 of size 10, 66 of size 11, 78 of size 12, 91 of size 13, 105 of size 14, 120 of size 15, 136 of size 16, and 153 of size 17, summing to 969. He explains that the number of triangles of size k is the sum of the first (n-k+1) natural numbers.

  16. 70:00 – 75:00 70:00-75:00

    The instructor shows a 18-level triangle and counts the triangles of different sizes: 1 of size 1, 3 of size 2, 6 of size 3, 10 of size 4, 15 of size 5, 21 of size 6, 28 of size 7, 36 of size 8, 45 of size 9, 55 of size 10, 66 of size 11, 78 of size 12, 91 of size 13, 105 of size 14, 120 of size 15, 136 of size 16, 153 of size 17, and 171 of size 18, summing to 1140. He explains that the number of triangles of size k is the sum of the first (n-k+1) natural numbers.

  17. 75:00 – 75:11 75:00-75:11

    The video concludes with a final screen that says "THANKS FOR WATCHING". The instructor has finished explaining the methods for counting triangles in various configurations and has provided a comprehensive set of formulas and examples for students to use in their exam preparation.

The video provides a structured and methodical approach to solving triangle counting problems. It begins by establishing a foundational formula, n(n+1)/2, for counting triangles in a simple configuration where lines are drawn from the apex to the base. The instructor then systematically builds upon this by introducing more complex patterns, such as horizontal divisions and a large grid of small triangles. For each new type of figure, he demonstrates a clear counting strategy, often using summation to find the total. The core of the lesson is the application of a general formula for counting triangles in a large grid: the number of triangles of size k is the sum of the first (n-k+1) natural numbers, where n is the number of base segments. This allows for a quick calculation of the total number of triangles in any such configuration. The video is designed to equip students with a reliable and efficient strategy for tackling these common exam questions.

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