2007 Alabama ARML TST Problems/Problem 11
Problem
In how many distinct ways can a rectangular 
grid be tiled with 17 non-overlapping 
rectangular tiles?
Solution
There are either 17 vertical tiles, 14 vertical and 3 horizontal, 11 vertical and 6 horizontal, etc. We can imagine the horizontal tiles blocks of 3 1*1 tiles. Thus, there are
![\[\dfrac{17!}{17!}+\dfrac{15!}{14!}+\dfrac{13!}{11!\cdot 2!}+\dfrac{11!}{8!\cdot 3!}+\dfrac{9!}{5!\cdot 4!}+\dfrac{7!}{2!\cdot 5!}\]](http://latex.artofproblemsolving.com/9/a/5/9a58d8a8d4d7078e50681d33efdc946240d9632d.png)
It isn't that such a pain to compute, so we do:
![\[1+15+78+165+126+21=\boxed{406}\]](http://latex.artofproblemsolving.com/8/8/9/889ef5fe36370dac14bb883ea5e8a7264d6f5a8a.png)
