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Three dice, each of which has its 6 sides numbered 1

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jjjinapinch Senior | Next Rank: 100 Posts Default Avatar
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Three dice, each of which has its 6 sides numbered 1

Post Mon Aug 07, 2017 9:10 am
Three dice, each of which has its 6 sides numbered 1 through 6, are tossed. The sum of the 3 numbers that are facing up is 12. Is at least 1 of these numbers 5?

(1) None of the 3 numbers that are facing up is divisible by 3
(2) Of the numbers that are facing up, 2, but not all 3, are equal

Official Guide question
Answer: C

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Post Fri Dec 08, 2017 7:31 am
jjjinapinch wrote:
Three dice, each of which has its 6 sides numbered 1 through 6, are tossed. The sum of the 3 numbers that are facing up is 12. Is at least 1 of these numbers 5?

(1) None of the 3 numbers that are facing up is divisible by 3
(2) Of the numbers that are facing up, 2, but not all 3, are equal
We are given that three dice (numbered 1 to 6) are tossed and that the sum of the three numbers facing up is 12. We need to determine whether one of these numbers is 5.

Statement One Alone:

None of the 3 numbers that are facing up is divisible by 3.

The information in statement one is not sufficient to answer the question. The numbers facing up could be 4, 4, and 4, and none of these numbers are 5. However, the numbers facing up could be 5, 5, and 2, and at least one of these numbers is 5. Statement one alone is not sufficient to answer the question. We can eliminate answer choices A and D.

Statement Two Alone:

Of the numbers that are facing up, 2, but not all 3, are equal.

The information in statement two is not sufficient to answer the question. The numbers facing up could be 3, 3, and 6, and none of these numbers are 5. However, the numbers facing up could be 5, 5, and 2, and at least one of these numbers is 5. Statement two alone is not sufficient to answer the question. We can eliminate answer choice B.

Statements One and Two Together:

Using the information from statements one and two, we can determine that the only way to obtain a sum of 12, when none of the numbers are divisible by 3 and when exactly two of the numbers are equal, is if the numbers facing up are 5, 5, and 2. Thus, at least one of the numbers is 5.

Answer: C

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Scott Woodbury-Stewart Founder and CEO

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Post Mon Aug 07, 2017 10:25 am
First, write out the possibilities of 3 dice that sum to 12, since the possibilities are finite:

6, 5, 1
6, 4, 2
6, 3, 3
5, 5, 2
5, 4, 3
4, 4, 4

Since order doesn't matter, these are the only combinations. The ones with at least one 5 are highlighted in red.

When evaluating the statements, see if you can find an example WITHOUT 5's as well as one with 5's.

(1) None of the 3 numbers that are facing up is divisible by 3
Which of our examples fit here?
5, 5, 2 --> "yes" answer to question
4, 4, 4 --> "no" answer to question
Insufficient.

(2) Of the numbers that are facing up, 2, but not all 3, are equal
Which of our examples fit here?
6, 3, 3 --> "no" answer to question
5, 5, 2 --> "yes" answer to question
Insufficient.

(1) & (2) Together:
Which of our examples fit BOTH statements?
5, 5, 2 --> this is the only example that fits both.
This gives us a definitive "yes" answer to the question.
Sufficient.

The answer is C.

Btw, this is DS #315 in OG 2017.

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EdM in Mind, Brain, and Education
Harvard Graduate School of Education


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