There are 10 spots in a parking lot arranged in a single row. Three cars are parked randomly but in such a way that no two cars are adjacent to each other. What is the probability that there are at least two empty spots between any two cars?

You can view this as counting integer solutions. x1,x2,x3,x4 representing the spacing. x1 is the spots before the first car, x2 between the first and the second, x3 between the second and third, x4 after the third.

For the no two cars adjacent the constraints are: x1,x4>=0, x2,x3>=1, x1+x2+x3+x4=7. For at least two spaces, x1,x4>=0, x2,x3>=2, x1+x2+x3+x4=7.

Then you can use stars-and-bars to count the solutions.

http://en.wikipedia.org/wiki/Stars_and_bars_%28combinatorics%29

 

Subtracting 1 from x2,x3 turns x1,x4>=0, x2,x3>=1, x1+x2+x3+x4=7 into x1,x2,x3,x4>=0, x1+x2+x3+x4=5, and there are C(5+3,3) ways.

Subtracting 2 from x2,x3 turns x1,x4>=0, x2,x3>=2, x1+x2+x3+x4=7 into x1,x2,x3,x4>=0,x1+x2+x3+x4=3, and there are C(3+3,3) ways.

C(3+3,3)/C(5+3,3) = 5/14