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I am super stuck on this one

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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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Feb 23, 2024

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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.

Feb 24, 2024