If you’ve ever played the game of Yahtzee, you’ll know that often times, the last line to be filled in is YAHTZEE. Let's say I want to calculate the probability of rolling Yahtzee on a single turn. To this end, suppose that after you roll five dice, you are allowed to select any of the five dice and roll them again. At which point, you may select any of the five dice and roll them for a third time. This would help with calculating the probability of rolling YAHTZEE after any of the three rolls. First, consider the situation where you have exactly i dice of the same value and you re-roll the other 5 − i dice. Assuming that you have i dice of the same value that you are not going to re–roll, let pi,j denote the probability that you end up with exactly j dice of the same value after you re–roll the other 5 − i dice. (please let me know what you think of this) Note that if j < i then pi,j = 0 since you certainly have at least i dice all of the same value after you roll the other 5 − i dice. I want to use a transitional matrix to calculate this which is easy with the binomial distribution, but let's say we wanted to calculate each value in the matrix using basic counting knowledge and probability theory.