Difference between revisions of "2007 Alabama ARML TST Problems/Problem 11"

(New page: ==Problem== In how many distinct ways can a rectangular <math>3\cdot 17</math> grid be tiled with 17 non-overlapping <math>1\cdot 3</math> rectangular tiles? ==Solution== There are eithe...)
 
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{{ARML box|year=2007|state=Alabama|num-b=10|num-a=12}}

Revision as of 12:56, 29 April 2008

Problem

In how many distinct ways can a rectangular $3\cdot 17$ grid be tiled with 17 non-overlapping $1\cdot 3$ 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

\[\dfrac{17!}{17!}+\dfrac{15!}{14!}+\dfrac{13!}{11!\cdot 2!}+\dfrac{11!}{8!\cdot 3!}+\dfrac{9!}{5!\cdot 4!}+\dfrac{7!}{2!\cdot 5!}\]

It isn't that such a pain to compute, so we do:

\[1+15+78+165+126+21=\boxed{406}\]

See also

2007 Alabama ARML TST (Problems)
Preceded by:
Problem 10
Followed by:
Problem 12
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