Solution: Given, there are 4 married couples Total number of people = 8 we have to find how many ways can 4 married couples be seated in a row a) if there are no restrictions on seating arrangement The number of possible ways = n! Here, n! = 8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40320 ways. Therefore, there are 40320 ways that the people can be seated when there is no restriction on the seating arrangement. b) if persons A and B must sit next to each other The possible ways = 7! × 2! = (7 × 6 × 5 × 4 × 3 × 2 × 1) × (2 × 1) = 5040 × 2 = 10080 Therefore, if persons A and B must sit next to each other there are 10080 ways of seating arrangement. c) if there are 4 men and 4 women and no 2 men or 2 women can sit next to each other This implies the restriction of having persons of opposite sex next to each other. We can have any one of the 8 persons in the first position. So, both man and woman can be arranged without having same next to each other = 8 × 4 × 3 × 3 × 2 × 2 × 1 × 1 = 1152 ways Therefore, if there are 4 men and 4 women and no 2 men or 2 women can sit next to each other, there are 1152 ways of seating arrangement. d) if there are 5 men and they must sit next to one another There are 5 men and 3 women. 5 men sit next to one another. The possible number of ways = 5! × 4! = (5 × 4 × 3 × 2 × 1) × (4 × 3 × 2 × 1) = 120 × 24 = 2880 ways Therefore, if there are 5 men and they must sit next to one another, there are 2880 ways of seating arrangement. e) if there are 4 married couples and each couple must sit together 4 pairs of couples can be arranged in 4! Each couple can be arranged in 2! Ways. The possible number of ways = 2! × 2! × 2! × 2! = (2× 1) × (2 × 1) × (2 × 1) × (2 × 1) × (4 × 3 × 2 × 1) = 2 × 2 × 2 × 2 × 24 = 16 × 24 = 384 ways Therefore, if there are 4 married couples sitting together, there are 384 ways of seating arrangement. In how many ways can 4 married couples (total of 8 people) be seated in a row if: (a) there are no restrictions on the seating arrangement? (b) persons A and B must sit next to each other?(c) there are 4 men and 4 women and no 2 men or 2 women can sit next to each other? (d) there are 5 men and they must sit next to one another? (e) there are 4 married couples and each couple must sit together?Summary: The possible number of ways can 4 married couples (total of 8 people) be seated in a row if: (a) there are no restrictions on the seating arrangement is 40320 ways (b) persons A and B must sit next to each other is 10080 ways (c) There are 4 men and 4 women and no 2 men or 2 women can sit next to each other in 1152 ways. (d) there are 5 men and they must sit next to one another is 2880 ways (e) there are 4 married couples and each couple must sit together in 384 ways.
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PHOTOGRAPHY A photographer is taking a picture of a bride and groom together with 6 attendants. How many ways can he arrange the 8 people in a row if the bride and groom stand in the middle? . - Introduction to Permutations - PROOF of the formula on the number of Permutations - Simple and simplest problems on permutations - Special type permutations problems - Problems on Permutations with restrictions - OVERVIEW of lessons on Permutations and Combinations in this site. Also, you have this free of charge online textbook in ALGEBRA-II in this site - ALGEBRA-II - YOUR ONLINE TEXTBOOK. The referred lessons are the part of this online textbook under the topic "Combinatorics: Combinations and permutations". Save the link to this textbook together with its description Free of charge online textbook in ALGEBRA-II https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson into your archive and use when it is needed.7. Six couples want to have their pictures taken. In how many ways can they arrange themselves in a row i
couples must stay together?
Given: n = r = 6 couples = 12 people Formula: P(n,r) = n!/(n-r)! Solution: P(12,12) = 12!/(12-12)! P(12,12) = 12! P(12,12) = 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 479,001,600 ways |