A quick puzzle this week.

Your friend offers you a bet: Roll six dice, and if you get *exactly four distinct numbers* you win.

**Question**: would you accept this deal at even odds?

For example a roll of 1,2,3,6,3,6 would win but a 1,6,3,4,2,6 would be a loss.

WIN

LOSS

Give it a few minutes thought …

Image: rekre89

There are many ways to solve this. One of these is with brute force. There are only 46,656 possible combinations the dice can land on (6 × 6 × 6 × 6 × 6 × 6), and it’s just a couple of lines of code to write some nested loops and perform the check.

Here is a table showing the frequency count of every distinct combination of the spots.

Distinct Count | Frequency |
---|---|

1 | 6 |

2 | 930 |

3 | 10,800 |

4 | 23,400 |

5 | 10,800 |

6 | 720 |

TOTAL | 46,656 |

As you can see, the total number of combinations with a distinct count of four is 23,400/46,656 which simplifies to 325/648, which is just a little more than 50:50, so it’s (just) slightly better than even odds. Yes, it’s a good bet to take!

*Yes, it's a good bet.*

Solving with brute force might not be very elegant, but it’s simple to write. You could optimize a little from symmetry, but let’s apply logic and come up for a formal solution.

There are only two possible ways we can get a distinct count of four:

Three of a kind, then three distinct singletons.

Two (different) pairs, with two distinct singletons.

There are 6 possible values for what the triplet dice would be, and there are *6 choose 3* ways this triplet could appear. There are then 5 values that the first singleton could be, 4 values for the second, and 3 for the last singleton. This is 6 × _{6}*C*_{3} × 5 × 4 × 3 = 7,200 possible combinations.

7,200/46,656 = 25/162.

With two pairs, there are _{6}*C*_{2} for what these numbers are. The first pair has _{6}*C*_{2} choices of where to go, and the second pair has _{4}*C*_{2} choices of position. There are then 4 possible combinations for the first singleton, and 3 possible for the last singleton.

This is _{6}*C*_{2} × _{6}*C*_{2} × _{4}*C*_{2} × 4 × 3 = 15 × 15 × 6 × 4 × 3 = 16,200 possible combinations.

16,200/46,656 = 25/72.

Adding these together, we get (7,200+16,200)/46,656 = 23,400/46,656 = 325/648

This confirms the answer we obtained from brute force! (Approx 50.15%)

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