Final answer:
The married couples can be seated next to each other in 48 ways, whereas the number of ways for the ladies to sit together is 144 ways in a row of six seats in a cinema hall.
Step-by-step explanation:
There are two parts to this question regarding seating arrangements. First, we'll address the scenario with married couples wanting to sit next to each other in a row of six seats in a cinema hall, and then we'll look at the situation where all the ladies sit together.
Married Couples Seating Together
Assume there are three married couples. When couples must sit together, we can consider each couple as a single unit. There are 3 units (couples) that can be arranged in 3! (factorial) ways. Within each unit, the two individuals (husband and wife) can switch places, resulting in 2 possible arrangements per couple.
So, the total number of ways for the couples to be seated next to each other is 3! × 2× 2× 2 = 48 ways.
Ladies Sitting Together
Assuming there are 3 ladies and they wish to sit together, we treat the group of ladies as one unit. Then, we have the group of ladies plus the remaining 3 seats for the men, totaling 4 units to arrange. These can be arranged in 4! ways. The ladies within their single unit can also be arranged in 3! ways.
Thus, the number of ways for the ladies to sit together is 4! × 3! = 144 ways.