Four married couples have bought 8 seats in row for a show. In how many different ways can they be seated.
A. If each couple is sit together?
B. If all the women sit together?
C. If all the women sit together to the right of all the men?
A. If each couple is sitting together, we can think of each couple as a unit. So we have 4 units to arrange. The number of ways to arrange these units is 4!, which is equal to 24. Within each couple, there are 2 ways for the partners to sit, so we multiply the number of ways to arrange the units by 2 for each couple. Therefore, the total number of ways for the couples to be seated together is 24 * 2 * 2 * 2 * 2 = 192.
B. If all the women sit together, we can think of the group of women as a single unit. So we have 5 units to arrange: the group of women and the 4 couples. The number of ways to arrange these units is 5!, which is equal to 120. Within each couple, there are 2 ways for the partners to sit, so we multiply the number of ways to arrange the units by 2 for each couple. Therefore, the total number of ways for all the women to sit together is 120 * 2 * 2 * 2 * 2 = 960.
C. If all the women sit together to the right of all the men, we can think of the group of women as a single unit and the group of men as a single unit. So we have 3 units to arrange: the group of women, the group of men, and the 4 couples. The number of ways to arrange these units is 3!, which is equal to 6. Within each couple, there are 2 ways for the partners to sit, so we multiply the number of ways to arrange the units by 2 for each couple. Therefore, the total number of ways for all the women to sit together to the right of all the men is 6 * 2 * 2 * 2 * 2 = 96.