When playing Yahtzee, you roll five regular six-sided dice. How many different outcomes
are possible from a single roll? The order of the dice does not matter. We will answer
this question using the stars and bars method of counting. To begin, because we have five identical
dice or five indistinguishable dice, we have five stars. And because there are six distinguishable
outcomes or six possible outcomes per die, we have five bars. The number of bars is always one less
than the number of distinguishable outcomes or distinguishable items. This is because if we
have six outcomes, we would have to have five switches to switch from the first outcome to
the sixth outcome. From here, every string we can form of length 10 with five stars and five bars,
represents one possible outcome when rolling five regular six-sided dice in the game of Yahtzee,
when the order of the dice doesn't matter. Let's take a look at three possible stars and bars
charts. For the first chart we start with a star, which indicates we assign the first outcome of one
to, let's just say, the first die. And then we have two bars, which represents two switches. And
therefore we switch from one to two, and then two to three. Then we have another star, and therefore
three becomes the outcome of the second die. We have another bar, and therefore we switch from
the outcome of three to the outcome of four. And then we have two stars. Four becomes the outcome
for the third and fourth dice. And then we have two more bars, which means we switch from four to
five, and then five to six. And then we have one more star. Six becomes the outcome of the fifth
die. Now remember the order here doesn't matter. The outcome is one one, one three, two fours, and
a six. Looking at the second chart, we start with a bar, which indicates we switch from one to two.
And then we have two stars, which indicates two becomes the outcome of the first and second dice.
Next we have three bars, which represents three switches. We switch from two to three,
three to four, and then four to five. Then we have three stars. Five becomes the outcome of the third
fourth and fifth dice. And then we have a bar, which represents switching from five to six.
But there are no more stars, and therefore we're done. The outcome is two twos and three fives,
which is called a full house in Yahtzee. And then finally for the next chart, we start with three
bars, which indicates three switches. We switch from one to two, two to three, and three to four.
Then we have five stars, which means four is the outcome for all five dice. And then we have
two bars, which represents switching from four to five and five to six. But again there are no more
stars. And therefore we're done. And this outcome is called Yahtzee, when we have five of the same
outcomes. So again the number of strings we can form with five stars and five bars
is equal to the number of possible outcomes when rolling five regular six-sided dice in
the game of Yahtzee. And the number of ways is equal to 10 choose five.
We form the combination by summing the stars and bars and choosing the number of bars.
This shouldn't be surprising, if we think about bit strings
and we think of the stars as zeros and the bars as ones. The number of bit strings of length ten with
five ones or a weight of five is ten choose five. And ten choose five is equal to ten factorial
divided by the product of ten minus 5 factorial and 5 factorial. Let's go and simplify by
hand. 10 factorial is equal to 10 times 9 times 8 times 7 times 6 times 5 factorial.
Notice in the denominator, we have two five factorials. Let's expand one of them. Five
factorial is equal to five times four times three times two times one. And we have times
another five factorial. And now simplifying, five factorial divided by itself simplifies to one.
In the denominator, we have three times two which is six. Six divided by six simplifies to
one. Four and eight share a common factor of four. There's one four in four and two fours in eight.
Five and ten share a common factor of five. There's one five and five and two fives in ten.
We're left with two times nine times two times seven, which is equal to 252.
There are 252 possible outcomes, when rolling five regular six-sided dice
in the game of Yahtzee, when the order of the dice doesn't matter. I hope you found this helpful.