# 2005 Alabama ARML TST Problems/Problem 1

## Problem

Two six-sided dice are constructed such that each face is equally likely to show up when rolled. The numbers on the faces of one of the dice are $1, 3, 4, 5, 6,\text{ and }8$. The numbers on the faces of the other die are $1, 2, 2, 3, 3,\text{ and }4$. Find the probability of rolling a sum of $9$ with these two dice.

## Solution

We use generating functions to represent the sum of the two dice rolls: $(x+x^3+x^4+x^5+x^6+x^8)(x+2x^2+2x^3+x^4)=$ $x^2(1+x^2+x^3+x^4+x^5+x^7)(1+x+x^2)(1+x)$

The coefficient of $x^9$, that is, the number of ways of rolling a sum of 9, is thus $(1+2+1)=4$, out of a total of $6^2$ possible two-roll combinations, for a probability of $\frac 19$.

Alternatively, just note the possible pairs which work: $(5, 4), (6, 3), (6, 3)$ and $(8, 1)$ are all possible combinations that give us a sum of $9$ (where we count $(6, 3)$ twice because there are two different $3$s to roll). Thus the probability of one of these outcomes is $\frac{4}{36} = \frac19$.