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Problem. In the game Yahtzee, a player rolls five fair six-sided dice, and gets a Yahtzee if all five dice show the same number. After the initial roll, the player gets two chances to reroll some of the dice. What is the probability that, on the initial roll, at least two of the dice show the same number? Express your answer as a common fraction.


I really don't get this problem at all; I am weak in combinatoics. Solutions?

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That'd be $1-P \space (\text{all dice show a different number})$

And $P \space (\text{all dice show a different number}) = \dfrac{{6\choose 5}5!}{6^5}$

As the numerator is the number of lists of 5 distinct integers, each of which is b/w $1$ and $6$ inclusive. This is relevant as each such list corresponds to a configuration of the 5 dice, also there are $6\choose 5$ ways to choose which 5 distinct numbers appear on the dice and $5!$ ways to assign these to the $5$ dice.

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  • $\begingroup$ Great job! Nice use of complementary counting. $\endgroup$
    – Fleccerd
    Aug 30, 2020 at 1:57

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