How many ways can 5 people be seated around a circular table if two of them insist on sitting beside?

Answer:

48

Step-by-step explanation:

There are 6 people and 2 need to be together. We can make the 2 count as 1 unit, and deal with it later. The problem temporarily becomes sitting 5 people on a round table.

To solve this we need to use the formula for circular permutation. The formula for circular permutation is given by (n-1)!(n−1)! where n is the number of objects. In this case, n = 5.

Substituting to the formula, we get

\begin{gathered}(5-1)!\\4!\\4*3*2\\24\end{gathered}

(5−1)!

4!

4∗3∗2

24

There are 24 ways to arrange 5 people in a round table.

However, we know that one of the units counts for the other 2 that insist on sitting together. They can arrange themselves in 2! = 2 ways (person one, person two; or person two, then person one)

We multiply this to the number of ways 5 people can be sit in a table; because of fundamental principles of counting. Imagine that for those 24, person one comes first, we haven't counted the times where person two comes before person one. Person two coming first before person one still counts because person one and person two are still sitting beside each other.

\begin{gathered}24*2!\\= 24*2*1\\=48\end{gathered}

24∗2!

=24∗2∗1

=48

There are 48 ways of sitting 6 people in a table with two of them insisting to be sitting beside each other.

. How many ways can 11 people be seated around a circular table if two of them insist to sit next to each other? ~~~~~~~~~~~~~~~ Then we consider this special pair as one (glued) object, and we, actually, have 10 objects then (instead of 11) to arrange around the circular table. We can arrange 10 objects around the circular table by 9! ways, but this special object can be in one of the two states (AB) or (BA). Therefore, the total number of all possible arrangements (circular permutations) of this kind is 2*9! = 2 * (9*8*7*6*5*4*3*2*1) = 725760. ANSWER Solved, answered and explained.

In order to continue enjoying our site, we ask that you confirm your identity as a human. Thank you very much for your cooperation.

$\begingroup$

I'm attempting this question, but I'm a little unsure about my answers. The full question is:

Five people are to be seated around a circular table. Two seatings are considered the same if one is a rotation of the other.

1. How many seatings are possible?
2. How many are possible if John and Mary always insist on sitting next to each other?
3. What is the probability if the five people are place randomly around the table that John and Mary will end up sitting next to one another?

My answers are:

1. (5-1)! = 4! = 24
2. 3! = 6
3. C(5,2) = 10