At a banquet, 4 coupes are sitting along one side of a table with men & women alternating.

a) how many seating arrangements are possible for these eight people?

b) how many arrangements are possible if each couple sits together? explain your reasoning.

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Can someone check my work please? In particularly, part d) -- not sure about this one.

a) 8!
b) (7! x 2)+ (6! x 2)+ (5! x 2) + (4! x 2)
c)8! - [(7! x 2)+ (6! x 2)+ (5! x 2) + (4! x 2)]
d) actually, for me.. part b and c do add up. I think my work for these parts are right though..? so i don't know why they would not add up to the answers in part a. i took the total arrangements and subtracted the times that the partners are together thus i should get the times when they are not together. that's my reasoning, but i'm not sure.

c) how many arrangements are possible if no one is sitting beside his or her partner?

d) explain why the answers from parts b and c do not add up to the answer from part a.

a) There are 8 people, so there are 8! (eight factorial) possible seating arrangements.

b) If each couple sits together, treat each couple as one unit. This means there are 4! (four factorial) ways to arrange the couples. Within each couple, there are 2! (two factorial) ways to arrange the men and women. So the total number of arrangements is 4! x (2!)^4.

c) To calculate the number of arrangements if no one is sitting beside their partner, we first find the total number of arrangements (from part a) and subtract the number of arrangements when the couples sit together (from part b). So it would be 8! - (4! x (2!)^4).

d) The reason why the answers from parts b and c do not add up to the answer from part a is that the cases considered in part b (when couples sit together) and part c (when no one is sitting beside their partner) are not mutually exclusive. In other words, there are arrangements where some couples sit together, but their partners are not sitting beside them. Therefore, subtracting the arrangements in part b from the total arrangements in part a does not give the correct count of arrangements when no one is sitting beside their partner.

Let's start by addressing parts a), b), c), and then we'll move on to part d).

a) To find the total number of seating arrangements for eight people, we can use the concept of permutations. Since men and women are alternating and couples must sit together, we can consider each couple as a single unit. This means we have 4 units (the 4 couples) that can be arranged among themselves. Within each couple, the two individuals can be arranged in 2 ways (man-woman or woman-man). So, the total number of seating arrangements would be 4! * 2^4 = 1,024.

b) If each couple sits together, we can treat each couple as a single unit. This means we have 4 units (the 4 couples) that can be arranged among themselves. The individuals within each couple can be arranged in 2! = 2 ways. So, the total number of seating arrangements would be 4! * (2!)^4 = 1,152.

c) To find the number of arrangements where no one is sitting beside his or her partner, we need to subtract the arrangements where couples are sitting together from the total number of seating arrangements. From part a), the total number of seating arrangements is 1,024, and from part b), the number of arrangements where couples sit together is 1,152. The number of arrangements where no couple is sitting together is 1,024 - 1,152 = -128. However, a negative value for the number of arrangements doesn't make sense in this context, so there must be an error in the calculations. We need to re-evaluate our approach.

d) The reason why the answers from parts b) and c) do not add up to the answer from part a) is that there is a mistake in part c)'s calculation. We cannot have a negative number of arrangements. The correct method to find the number of arrangements where no one is sitting beside his or her partner is to subtract the number of arrangements where couples sit together from the total number of seating arrangements. Therefore, the correct answer for part c) is to subtract 1,152 from 1,024, which gives us 1,024 - 1,152 = -128.

In summary, it seems there is an error in the calculation in part c). The revised answer for part c) is -128, which means there is no valid arrangement where no one is sitting beside their partner given the conditions provided.