Short Tricks to Solve Dice Sum Problem (1)

Duration: 10 min

This video lesson is available to enrolled students.

Inside: a video lesson and guided study material.

Module outline

Paper 1 | Reasoning Test: Series (Number and Letter Series) (Numerical Relations and Reasoning), Coding Decoding, Data Sufficiency, Syllogisms, Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Seating Arrangements, Blood Relations, Puzzle Solving (Puzzles), Venn Diagram, Analogy, Ranking, Order and Sequence, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Selection Decision Table (Decision Making), Logical Sequence Of Words, Cubes & Dices, Analytical Reasoning (Counting Figures Reasoning), Calendar, Alphabet and Word Tests, Numerical Reasoning, Number System, Averages, Mixtures and Alligation, Percentages, Series Completion, Analogy and Classification, Patterns in Two-Dimensional Figures, Coding and Decoding, Mathematical Operations, Direction Sense, Algebra, Logical Deduction and Induction, Syllogisms and Venn Diagrams, Critical Reasoning, Order, Ranking and Sequence, Reading Comprehension, Profit, Loss and Discount, Constraint Puzzles, Speed, Time and Distance, Traverse Surveying, Ratio and Proportion, Time and Work, Mensuration, Permutations and Combinations, Clocks and Calendars, Data Interpretation, Assembling and Grouping of Shapes, Transformation of Shapes, Three-Dimensional Patterns and Visualization, Cubes and Dice, Vocabulary, Blood Relations, Data Sufficiency
Paper 1 | Numerical Analysis: Number System, HCF LCM, Percentage, Ratio and Proportion (Ratios), Average, Sequence and Series (Series), Powers and Exponents (Surds and Indices), Algebra, Statistics, Data Interpretation, Mental Ability, Arithmetic, Logarithm, Profit and Loss, Simple Interest and Compound Interest, Speed Time and Distance, Mixture and Alligation, Mensuration and Geometry, Permutation and Combination, Probability, Sequences and Series, Powers, Surds and Logarithms, Subject-Verb Agreement, Sentence Completion, Geometry, Sentence Structure and Correction, Data Sufficiency, Trigonometry, Measurement and Quantity Take-Off, Signal Space and Optimum Detection, Probability, Statistics, Algebra, Time and Work, Speed, Time and Distance, Data Interpretation, Graph Theory, Combinatorics
Paper 1 | General Knowledge: State Wise GK, History, Geography, GK- Current Affairs, Static General Knowledge, Polity, Indian Economy
Paper 1 | DBMS: Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, File Organization & Indexing, Relational Algebra, SQL, Transaction Management, Concurrency Control, Relational Calculus, Database Security & Authorization, Database Recovery
Paper 1 | Data Communication & Network: 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, Network Security, Security in Mobile and Wireless Computing, Input Output Organisation, File Management, Hardware Basics, Terms & Abbreviations, Protocols & Services, Network Access Technologies, Cyber Safety and Security, Basics of COA
Paper 2 | System Analysis & Design: Fundamentals of Systems and Information Systems, SDLC and Development Approaches, Fundamentals of Software Engineering, Software Requirements and Quality Assurance, Software Documentation and Software Requirements Specification (SRS), SDD and Quality Assurance, Software Design, Estimation and Metrics, Software Testing, Software Maintenance and Configuration Management, Security, Risk Management & Maintenance, extras
Paper 2 | Programming Language (Java): Java Fundamentals, OOPS with Java, Array, String, IO & References, Generic & Collections, Multithreading, Java, Miscellaneous, Control Flow, OOPS with C++, Stack, C Fundamentals, Arrays & Pointers, Object Oriented Technology, Control Flow & Looping, Introduction to Python

AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This educational video is a mathematics lecture that teaches how to solve problems involving the number of ways dice can sum to a specific value. The instructor begins by introducing the concept of permutations and combinations, using a visual metaphor of a detective trying to unlock a combination lock to explain the difference between the two. The main focus is on a 'Dice Sum Problem' where the goal is to find the number of ways three dice can sum to a given number. The video demonstrates a method using the stars and bars technique, which involves transforming the problem into finding the number of non-negative integer solutions to an equation. The instructor shows the general formula for this, n+r-1Cr, and applies it to the specific case of finding the number of ways to get a sum of 6 with three dice. The solution is derived by setting up the equation x1 + x2 + x3 = 6, where each xi represents the value on a die, and then applying the formula with n=6 and r=3. The video concludes with a similar example for a sum of 5, reinforcing the method. The lecture is presented on a digital whiteboard with the instructor visible in a small window, and the content is from 'Knowledge Gate Eduventures'.

Chapters

0:00 – 2:00 00:00-02:00
The video opens with a title card that visually distinguishes between 'PERMUTATION' and 'COMBINATION' using a casino theme (dice and roulette) and a business theme (fruit basket and office). This is followed by a slide titled 'Confused' which poses the question, 'Should I unlock with Permutation or Combination?'. The slide uses a detective metaphor, showing two locked doors labeled 'Permutation' and 'Combination', to illustrate the difference. The instructor, Yash Jain, is visible in a small window, and the 'Knowledge Gate Educator' logo is present. The video then transitions to a new slide titled 'Dice Sum Problem' with an image of two dice in a hand, setting up the main topic of the lecture.
2:00 – 5:00 02:00-05:00
The instructor presents the first problem: '3 dice are thrown, number of ways they can sum 6?'. He begins to solve it by setting up the equation x1 + x2 + x3 = 6, where x1, x2, and x3 are the values on the three dice. He notes that each die must show a value of at least 1, so x1 ≥ 1, x2 ≥ 1, x3 ≥ 1. To apply the standard formula for non-negative integer solutions, he introduces a substitution: x1' = x1 - 1, x2' = x2 - 1, x3' = x3 - 1. This transforms the equation into x1' + x2' + x3' = 3, where x1', x2', x3' ≥ 0. He then applies the formula for the number of non-negative integer solutions, which is n+r-1Cr, where n is the sum (3) and r is the number of variables (3). He writes the formula as 3+3-1C3, which simplifies to 5C3.
5:00 – 10:00 05:00-10:00
The instructor continues the calculation for the number of ways to get a sum of 6. He writes the formula 5C3 and then calculates it as 5! / (3! * 2!) = (5 * 4) / (2 * 1) = 10. He confirms that there are 10 ways. He then moves to a second example: 'If three dice are thrown, find the number of ways to get sum 5?'. He sets up the equation x1 + x2 + x3 = 5 with x1 ≥ 1, x2 ≥ 1, x3 ≥ 1. He applies the same substitution method, transforming it to x1' + x2' + x3' = 2. He then applies the formula n+r-1Cr with n=2 and r=3, which is 2+3-1C3 = 4C3. He calculates 4C3 as 4! / (3! * 1!) = 4. He concludes that there are 4 ways to get a sum of 5.
10:00 – 10:13 10:00-10:13
The video concludes with a black screen displaying the text 'THANKS FOR WATCHING' in white and orange. This is a standard closing slide for the educational content.

The video provides a clear, step-by-step tutorial on solving dice sum problems using combinatorics. It begins by establishing the foundational concept of permutations versus combinations, using a relatable metaphor. The core of the lesson is the application of the stars and bars method to find the number of integer solutions to an equation. The instructor demonstrates a systematic approach: first, setting up the equation for the sum, then adjusting for the constraint that each die must show a value of at least 1 by using a substitution, and finally applying the formula for combinations with repetition. The method is effectively demonstrated with two examples, first for a sum of 6 and then for a sum of 5, reinforcing the technique. The progression from a general problem to a specific, solved example makes the concept accessible for students preparing for competitive exams.