AIME I 2009 #9

[stupidityismygam]

A game show offers a contestant three prizes [unparseable or potentially dangerous latex formula]and [unparseable or potentially dangerous latex formula] each of which is worth a whole number of dollars from [unparseable or potentially dangerous latex formula] to [unparseable or potentially dangerous latex formula] inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order [unparseable or potentially dangerous latex formula] As a hint, the digits of the three prices are given. On a particular day, the digits given were [unparseable or potentially dangerous latex formula] Find the total number of possible guesses for all three prizes consistent with the hint.

[Celsion]

Well, let's start with A+B+C=7. A is the number of digits of prize A, B is the number of digits of prize B, C is the number of digits of prize C, and 7 is the total pool of the digits of the prizes. The price of the prizes can be as low as 1 digit or as high as 4 digits.

I think you can apply balls and bins. >_<
[unparseable or potentially dangerous latex formula], there are 15 groups, but this include the possibility of prizes with 5 digits, so you can eliminate the possibility of having 1,1,5; 1,5,1; and 5,1,1. That leaves 12 groups.

Now to determine the ways you can arrange the 1's and the 3's. [unparseable or potentially dangerous latex formula].

So, the total number of possibly guesses for all three prizes is [unparseable or potentially dangerous latex formula].