In how many ways can a family of six be seated at a round table if the mother and father must sit together?

treat the "mother and father" as one entity.

So we are arranging 5 around the table
which is 5!
but we can rotate each particular arrangement 5 times around the table without changing any order, since there is no specified position around the table
(e.g. everybody get up and move one seat to the right, the seating order is still the same. We can do this 5 times)

number of ways = 5!/5 = 4!
BUT, we could sit Mother-Father or Father-Mother, so
number of ways = 2x4! = 48

Well, if the mother and father must sit together, we can imagine them as a package deal. So, instead of considering them as two separate entities, we can consider them as one super-parent! That leaves us with five "individuals" to seat around the table.

Now, there are 5 factorial (5!) ways to arrange these five individuals. However, since the table is round, we need to account for the fact that rotations of the same arrangement are considered the same.

So, to answer your question, there are 5 factorial divided by 5 (5!/5) ways to seat the family of six at a round table if the mother and father must sit together. But let's be honest, they're probably going to argue over who gets to sit in the comfiest chair anyway!

To calculate the number of ways a family of six can be seated at a round table with the condition that the mother and father sit together, we can treat them as a single unit. This reduces the problem to seating five people at a round table.

Step 1: Fix the mother and father's position
Since the mother and father must sit together, we can think of them as a single unit. We fix their position at one of the seats.
There are 5! (5 factorial) ways to arrange the remaining five people around the table.

Step 2: Account for the arrangements within the mother and father unit
Since the mother and father can switch positions within their unit, we need to multiply the answer from step 1 by 2.

Step 3: Calculate the total number of ways
The total number of ways to seat the family is given by the product of the results from steps 1 and 2:
Total Ways = 5! * 2

Applying the formula:

Total Ways = 5! * 2
= 120 * 2
= 240

Therefore, there are 240 ways to seat a family of six at a round table if the mother and father must sit together.

To determine the number of ways the family of six can be seated at a round table if the mother and father must sit together, we can treat the mother and father as a single entity.

First, let's arrange the mother and father together. We can treat them as a single unit, which means there are 2! (2 factorial) ways to arrange them amongst themselves.

Next, we need to arrange the remaining four family members (considering the mother and father as one entity) and the empty seats around the table. There are (6 - 1) = 5 seats remaining.

The four family members can be arranged in (4!) = 4 * 3 * 2 * 1 = 24 ways.

Lastly, since the table is round, we need to divide the total number of arrangements by the number of seats. In this case, it would be 6 (total seats around the table).

So, the total number of ways the family can be seated is (2!) * (4!) / 6.

Simplifying this expression, we get: (2 * 1) * (4 * 3 * 2 * 1) / 6 = 2 * 24 / 6 = 48 / 6 = 8.

Hence, there are 8 ways in which the family of six can be seated at a round table if the mother and father must sit together.

48