AIME I #9 2008

[stupidityismygam]

Ten identical crates each have dimensions [unparseable or potentially dangerous latex formula] ft x [unparseable or potentially dangerous latex formula] ft x [unparseable or potentially dangerous latex formula] ft. The first crate is placed flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen at random. Let [unparseable or potentially dangerous latex formula] be the probability that the stack of crates is exactly [unparseable or potentially dangerous latex formula] ft tall, where [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are relatively prime positive integers. Find [unparseable or potentially dangerous latex formula].

[Roel]

[unparseable or potentially dangerous latex formula] # of 3 ft crates
[unparseable or potentially dangerous latex formula] # of 4 ft crates
[unparseable or potentially dangerous latex formula] # of 6 ft crates

[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
If we multiply the second equation and subtract them we get the Diophantine equation:
[unparseable or potentially dangerous latex formula]
This will give us the solutions [unparseable or potentially dangerous latex formula]
If we plug back this values to one of our equations:
[unparseable or potentially dangerous latex formula]
Now we have to figure out how many different ways these can be arragened. For our first triplet we do the following:
[unparseable or potentially dangerous latex formula]
And similarly for both of the other triplets:3
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula] different arrangements. Now we can easily see that the total number of possible arrangements is [unparseable or potentially dangerous latex formula]. The probability of having a stack 41 ft tall is [unparseable or potentially dangerous latex formula].
[unparseable or potentially dangerous latex formula]