There are 18 chairs numbered from 1-18 through around a circular table. How many ways can three people be seated, so that no two people are adjacent?
Anty the ant starts at the point (0,0) in the coordinate plane. At each step, Anty takes a step to the right, left, up, or down. After 10 steps, how many different points could Anty end up at?
1) There are 18*17*16 ways to choose the three seats, if people can be adjacent. We must subtract the cases where there are two people adjacent, which is 18*13*6. So the number of ways is 18*17*16 - 18*13*6 = 3492.
2) Anty can get to any point inside the square with vertices (5,5), (5,-5), (-5,-5), (-5,5), so there are are a total of 121 points.
