In how many ways can a family of seven sit round a table if the mother and father must sit together?
Treat the Father-Mother as one person, so now you are seating only 6.
Let's put MF in one of the seats, so with caring about the actual place around the table there would be 6*5! ways to do that.
BUT, supposedly there are no distinct seats, so the number of ways is 6*5!/6 = 5!
(image everybody getting up and moving one seat to the left, no new arrangement is created, but we could do this 6 times, therefore I divided by 6)
BUT, we could have seated father and mother as FM or MF, so the
number of ways to arrange the family is 2*5!
To determine the number of ways a family of seven can sit around a table while the mother and father must sit together, we can treat the couple (mother and father) as a single entity. This means we consider the couple as one unit and arrange the other five family members and the couple around the table.
First, let's consider the couple as a single entity. We can arrange this entity and the other five family members around the table in (6-1)! = 5! = 120 ways.
Next, within the couple, the mother and father can be arranged in 2 different ways (mother-father or father-mother).
To calculate the total number of arrangements, we multiply the number of ways to arrange the couple (2 ways) by the total number of ways to arrange the other family members (120 ways).
Therefore, the total number of ways a family of seven can sit around a table with the mother and father sitting together is 2 × 120 = 240 ways.