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The faces of two regular dodecahedra are labeled with the numbers 1 to 12 in order to make dice. If these dice are rolled, what is the probability that the sum of the two top numbers is greater than 16?

 Jun 22, 2022
 #1
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There are 12 x 12 = 144 ways to roll the 2 numbers. 

 

Let's count the amount of ways as shown: 

 

max digit = 12: (12, 5 - 12) = 8 ways 

max digit = 11: (11, 6 - 11) = 6 ways

max digit = 10: (10, 7 - 10) = 4 ways 

max digit = 9: (10, 8 - 9) = 2 ways 

 

There are 8+6+4+2 = 20 ways, which makes for a probability of \({20 \over 144} = \color{brown}\boxed{ 5 \over 36 } \)

 Jun 22, 2022
 #2
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 N=2; D=12;a=(listfor(S,S=17, S=24, (sumfor(k, 0, ((S - N)/D), ((-1)^k * (N nCr k) * (S - D*k - 1) nCr (N - 1));printa, ">>Total =",sum(a)

 

 

(8, 7, 6, 5, 4, 3, 2, 1) >>Total = 36

 

 

The probability is: 36 / 144 ==1 / 4

 Jun 22, 2022

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