2. In how many ways can 8 people be seated in a row if (a) there are no restrict
ID: 3355340 • Letter: 2
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2. In how many ways can 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 each other; (e) there 4 married couples and each must sit together? Solution: (a) Any person can sit any seats. There are thus 8! possibilities (b) First consider A and B as one person, arrange seating for 7 people without any restriction. Then permute A and B for every possible arrangement. The answer is 7!2! (c) Similar to (b), pair one man and one woman as one person Arrange seating for 4 virtual people first to get 4!. There are also 4! unique ways to pair a man and a woman, similar to arranging 4 men in 4 seats. There are two alternative man and woman seating for every arrangement and pairing, man on left or man or right. Total the answer is 4!412 (d) Group 5 men as one person, allowing 4! for these "virtual people". The permuation of 5 men is 5!. The answer is 4!5! (e) First we arrange seating for couples, to get 4!. Then, for each couple and each seating, there are 2 arrangements, man on the left or the right. (Note that it is allowed for two womern or two men to sit together in this case). The answer is 4121Explanation / Answer
(a) There are no restrictions on the seating arrangments means any person can sit on any seat. So, for the first person there are total 8 possible seats, now when the second person comes he has only 7 possible seats to sit on and this process will continue till the eight will sit. Since any person can sit on any sit, so total number of solutions will be 8*7*6*.....*1=8!
(b) Consider A and B as a single person(as they will sit next to each other only), now there are total of 7 members. So, the possibility for them is 7!. Now, A and B can interchange there positions. Then, there are 2 possible solutions for that (which can also be presented as 2!). For every 7! seating arrangement they can change interchange it. Hence, the total possible solutions will be 7!*2!.
(c) As no pair of men and women can sit together. Then, it is obvious a man should be seated next to a woman and vice versa. So, we can consider 1 man and 1 woman as a single person and do the same for the other 3 men and 3 women. Now, there are total of 4 person. So, the outcome will be 4!. We also have to consider the fact that any man can sit with any woman, that is why we will also calculate the possibility of pairing while seating them which is 4! (first man can choose any of the 4 ladies and so this selection is similar to one of selecting the seat). Now, no 2 men or no 2 women of different pair can sit by each other. So, we will decide sides either woman can always sit on the right side or vice versa which is 2!. Now, by this the total outcome will be 4!4!2!.
(d) Consider the five men as a single virtual person. Then there are 4 person in total. So, the outcome will be 4!. Now, 5 men can also change seats with each other whose possible outcome will be 5! (calculation similar to the one described in part a). Thus, the total possible seating are 4!5!
(e) Married couple will sit next to each other and there are 4 couples, which gives 4!. Now each couple can interchange there seats(there is no restriction of 2 women or 2 men sitting together here) which give 24 ( calculation similar to the one given in part (b) for A and B, there are 4 couples so we multiply it 4 times). Now, the total is 4!24.
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