2008 AMC 12B Problems/Problem 22
Problem
A parking lot has 16 spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers chose spaces at random from among the available spaces. Auntie Em then arrives in her SUV, which requires 2 adjacent spaces. What is the probability that she is able to park?
Solution
Auntie Em won't be able to park only when none of the four available spots touch. We can form a bijection between all such cases and the number of ways to pick four spots out of 13: since none of the spots touch, remove a spot from between each of the cars. From the other direction, given four spots out of 13, simply add a spot between each.
So the probability she can park is ![\[1-\frac{{13 \choose 4}}{{16 \choose 4}}=1-\frac{13\cdot12\cdot11\cdot10}{16\cdot15\cdot14\cdot13}=1-\frac{11}{28}={\textbf{(E)}\frac{17}{28}}.\]](http://latex.artofproblemsolving.com/5/8/d/58d8d8218d347f2f680c6e9126a441c5d42abdfb.png)
(Bijection: When elements of two sets are perfectly paired with each other, i.e. each and every element from both sets has exactly one match in the other, and no elements are left out. In the context of this problem, this means the number of distinct ways to order the cars such that no two spaces are adjacent is exactly the number of ways to pick 4 spots out of 13.)
Further Explanation
Instead of trying to park the cars one at a time(considering the number of choices for the first car, then the second, and so on), imagine that all of the cars are already parked, and ignore the empty parking spaces for now. This will take up 
parking spaces, illustrated as asterisks:
![\[************\]](http://latex.artofproblemsolving.com/8/6/8/868f038309fd7816f7f6316e873398907e4b8b65.png)
We know that 4 spaces must be left empty, and we care about the arrangements where no empty spaces are next to one another. Notice that this is simply the number of ways to put 
|s into 
slots (including the "slot" in front of the first car as well):
![\[|*****|*|*|***** \text{(this works)}\]](http://latex.artofproblemsolving.com/8/a/0/8a03318c69fabe6888e9b8fb72f9ee86abf82ae6.png)
![\[|******||*|***** \text{(this doesn't)}\]](http://latex.artofproblemsolving.com/9/b/a/9ba803db41a96aaddcc5cad199c807e504cd96ff.png)
In other words, after a space between two cars has been selected to be empty, that space is no longer an option for the next selection. The number of available spaces will decrease by one every time, so we have 
ways to arrange the empty spaces. But the empty spaces are identical, so we must divide by the number of orders in which we can place the empty spaces, or 
. This is equivalent to 
total ways to arrange empty spaces into "slots". Then we continue similarly to the above.
Note
A much quicker way to think about it is to simply realize that the person needs 
spaces. Thus, we can consider the 
spaces we have in the beginning and literally subtract those two imaginary spaces wherever they might be. Once that is done, we are assuming that those two adjacent spaces have been taken care of thus resulting in choose ways. I'm not sure if this really works or if it is just a coincidence.
Who tf wrote this there are 16 original spaces so why would this work.
See Also
| 2008 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 21 |
Followed by Problem 23 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
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