There are $18$ padded chairs around a circular table, and the chairs are numbered from $1$ through $18$. How many ways can three people take their seats, so that no two people are adjacent?

The total number of ways = 3276

Step-by-step explanation:

When there is no restriction, there a total of ways the three-person can sit, 18*17*16 = 4896 ways

The ways to choose or decide 3 consecutive seats = 18

So three people can be placed in 3 = 6 ways.

So total possible seating 18 * 6 = 108

Now, The ways to choose or decide 2 consecutive seats = 18

So there are 14 ways to choose the third seat. So ways to place three people 18*14*6 = 1512

The number of ways in which 3 people take a seat and no 2 people are adjacent, 4896-108-1512 = 3276