Findthenumberofways6boys6girlsinarow.(i)allthegirls(ii)notwo girlsaretogether(iii)both alternately ? скачать в хорошем качестве

Findthenumberofways6boys6girlsinarow.(i)allthegirls(ii)notwo girlsaretogether(iii)both alternately ?

Find the number of ways of arranging 6 boys and 6 girls in a row. In how many of these arrangements. (i) all the girls are together. (ii) no two girls are together (iii) boys and girls come alternately ? Introduction: Welcome to this video where we will explore the fascinating topic of permutations and combinations. We will specifically look at the problem of arranging a group of 6 boys and 6 girls in different orders. We will also examine the number of ways in which the arrangement can be made under different conditions. In this video, we will consider three scenarios: (i) all the girls are together (ii) no two girls are together (iii) boys and girls come alternately Before we dive into the problem, let's review some basic concepts of permutations and combinations. Permutations: A permutation is an arrangement of objects in a specific order. When we arrange objects, we are essentially creating different sequences. The number of sequences that can be formed from a set of n objects is given by n factorial, denoted as n!. For example, if we have 3 objects, the number of sequences that can be formed is 3! = 3 x 2 x 1 = 6. The formula for the number of permutations of n objects taken r at a time is given by: nPr = n!/(n-r)! Combinations: A combination is a selection of objects without regard to order. The number of combinations that can be formed from a set of n objects taken r at a time is given by the formula: nCr = n!/r!(n-r)! Problem: Now, let's consider the problem of arranging 6 boys and 6 girls in a row. We will look at the number of ways in which the arrangement can be made under different conditions. (i) All the girls are together: In this scenario, we need to arrange the 6 girls in a row. This can be done in 6! ways. Once the girls are arranged, we can arrange the 6 boys in the remaining 7 positions in 6! ways. Therefore, the total number of arrangements where all the girls are together is: 6! x 6! = 518400 (ii) No two girls are together: In this scenario, we need to arrange the 6 boys and 6 girls in such a way that no two girls are together. One way to approach this problem is to use the principle of inclusion-exclusion. We can start by arranging the 6 boys in a row in 6! ways. We can then consider the cases where one, two, three, four, five, or all six girls are together. If one girl is together, we can treat that group of girls as a single object and arrange the 5 objects (4 groups of boys and 1 group of girls) in 5! ways. There are 6 ways to choose the group of girls, and once they are chosen, there are 5! ways to arrange the objects. Therefore, the number of arrangements where one girl is together is: 6 x 5! x 6! = 3110400 If two girls are together, we can treat each group of girls as a single object and arrange the 4 objects (3 groups of boys and 2 groups of girls) in 4! ways. There are 5 ways to choose the pair of girls, and once they are chosen, there are 4! ways to arrange the objects. However, we have overcounted the cases where both pairs of girls are together. There are 6 ways to choose the pairs of girls, and once they are chosen, there are 3! ways to arrange the objects. Therefore, the number of arrangements where two girls are together is: 5 x 4! x 6! - 6 x 3! x 6! = 2764800 If three girls are together, we can treat each group of girls as JOIN LIVE CLASSES https://chat.whatsapp.com/GmV7W51d8LT... website : www.e-math.in contact : +91 7032507699(whatsapp) visit my website: www.e-math.in youtubechannel:    / onlinemathstutionguntur   facebook page:   / emath2016   twitter :  / e-math2016   whatsapp :+91 7032507699 e-mail : e-math2016@outlook.com #Permutations #Combinations #Mathematics #Probability #Arrangements #BoysAndGirls #InclusionExclusionPrinciple #Counting #MathProblems #MathHelp #STEM #Education #LearnMath #YouTubeTutorial