Question

There are 7! = 5040 ways in which seven people can form a line. In how many

ways can seven people form a circle

Answer 1

Let us find the number of ways that seven people form a circle. Let us fix some position. For this position thare are 7 ways to seat a person. Let us move in the clockwise direction. For the next position thare are 6 ways to seat a person, and so on. For the last postion there is one way to seat a person. By Combinatirial Product Principle, the number of ways that seven people form a circle with fixed initial position on the circle is equal to $7⋅6⋅5⋅…⋅1=5040.$ Since each way with fixed position generates 7 different ways that form the same circle, the number of ways that seven people form a circle is equal to $\frac{5040}7=720.$