Multinomial Coefficients and Farkle

This is the first in a series of planned articles about Farkle and its strategy.  Lately, I’ve been spending alittle too much time playing Farkle on Facebook. In case you’re not familiar, Farkle is a dice game. It used to be pretty well-known before becoming overshadowed by the popularity of its distant cousin Yahtzee.

Like all the old games, there are a certain number of variations, but the Facebook version is pretty mainstream. For the purposes of this article, we’ll use that version of the rules.

The goal is to gain points by rolling scoring combinations with dice. The scoring combinations are as follows: three (or more) of a kind, 1’s and 5’s, three pairs, and a 1-2-3-4-5-6 straight. (For today’s purposes we don’t care how much the various scoring combinations are worth, only what they are.) You start with six dice. After each roll, you set aside one or more scoring combinations (you must score at least some dice after each roll). Then you can either roll the remaining dice, hoping for more points, or save your points and end the turn.  If you use up all six dice in scoring combinations, you get a new set of six dice.  But if ever you roll and you don’t get any scoring combinations, you “farkle out” and lose all the points you’ve scored this turn. (This is a bad thing.)

Today we’ll start with a fundamental question: if I roll a certain number of dice, how likely am I to farkle out?