Video Transcript
Determine the number of ways six children can sit in a circle.
Let’s consider the circle as shown with six chairs placed on its circumference. We might think that as there are six children, the total number of ways they can sit would be equal to six factorial. This is not the case however. If we consider the six children 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, and 𝐹, they can sit as shown in the diagram.
Let’s now assume that each child moves one place in the clockwise direction. They will now be sat in the seats as shown. As we are not told where the circle starts, this order is exactly the same as the previous one. This means it does not matter where the first child sits. Wherever child 𝐴 sits, the second child 𝐵 will have five possible options. Once they have chosen a seat, child 𝐶 will have four possible options. Child 𝐷 will be left with three possible options. After they have chosen, child 𝐸 will have two possible options. And there will be one seat remaining for child 𝐹.
Using the fundamental counting principle, we can then multiply five, four, three, two, and one to calculate the total number of ways six children can sit in a circle. This can also be written as five factorial, which is equal to 120.
There are 120 ways that six children can sit in a circle.
