Two fair dice, each with at least 6 faces, are rolled. On each face of each die is printed a distinct integer from 1 to the number of 3 of the probability of rolling a sum of 10, and the probability 4 faces on that die, inclusive. The probability of rolling a sum of 7 is 1 What is the least possible number of faces on the two dice combined? 12 of rolling a sum of 12 is (А) 16 (В) 17 (C) 18 (D) 19 (E) 20
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The multiple-choice question can be solved by the method of elimination.
(A) First the option of 16 is considered.
Here, the two dice can have 10 and 6, 9 and 7, or 8 and 8 number of faces, respectively.
Now, the information that probability of getting a sum of 12 is is considered.
If number of faces are 10 and 6,
Total number of possible outcomes are= 10x6= 60
Possible outcomes for getting 12 are {(10,2), (9,3), (8,4), (7,5), (6,6)}
So, probability of getting 12 is .
It is observed here that the given information holds.
If number of faces are 9 and 7,
Total number of possible outcomes are= 9x7= 63
Possible outcomes for getting 12 are {(9,3), (8,4), (7,5), (6,6), (5,7)}
So, probability of getting 12 is . Hence, the given information does not hold.
If number of faces are 8 and 8,
Total number of possible outcomes are= 8x8= 64
Possible outcomes for getting 12 are {(8,4), (7,5), (6,6), (5,7), (4,8)}
So, probability of getting 12 is . Hence, the given information does not hold.
So, the information holds when the two dice have 10 and 6 faces respectively.
Therefore, option (A) is the answer, that is, 16 is the least possible number of faces on the two dice combined.
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