+0 In the dice game Zero, a player rolls six standard six-sided dice and then scores points for certain combinations, as shown in the table.
A die cannot be used in more than one combination. For example, a player who rolls 156446 scores 50 points, while a player who rolls 144414 scores 250 points. What is the probability that a player scores 0 points on a roll? Express your answer as a common fraction.
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+1768 We can solve this problem by counting the number of rolls that don't score any points and then dividing by the total number of possible rolls. Here's how:
Rolls that score 0 points:
Six singles: There are 5 ways to roll a single 1, 5 ways to roll a single 2, etc., giving 6⋅5=30 ways to roll six singles.
Three pairs: There are (26)=15 ways to choose two dice for the first pair, and then 4 ways to choose a value for that pair (since the dice cannot be used in more than one combination).
Similarly, there are 4 choices for the second pair, giving a total of 15⋅4⋅4=240 ways to roll three pairs.
One triple and one pair: There are (16)=6 ways to choose the die for the triple, and then 5 choices for its value.
There are (25)=10 ways to choose the two dice for the pair, and then 4 choices for its value, giving a total of 6⋅5⋅10⋅4=1200 ways to roll one triple and one pair.
Four singles and one double: There are (16)=6 ways to choose the die for the double, and then 5 choices for its value.
There are (45)=5 ways to choose the four dice for the singles, giving a total of 6⋅5⋅5=150 ways to roll four singles and one double.
Total number of rolls:
There are 6⋅6⋅6⋅6⋅6⋅6 = 6^6 possible outcomes when rolling six dice.
Probability of scoring 0 points:
The probability of scoring 0 points is the number of rolls with no points divided by the total number of rolls: (30 + 240 + 1200 + 150)/^6 = 15/424.
