What is the Probability of Getting a Sum of 12 when Rolling 3 Dice GMAT Problem Solving

Question: What is the probability of getting a sum of 12 when rolling 3 dice simultaneously?

1. 10/216
2. 12/216
3. 21/216
4. 23/216
5. 25/216

Correct Answer: E
Solution with Explanation
Approach Solution 1:

let the dices be a,b,c
a+b+c = 12
but 0 < a,b,c < 6

so we can re-write the equation as
(6-a)+(6-b)+(6-c) = 12
=> a+b+c = 6

which contains a total of 28 C(8,2) whole number solutions; however, three of them must be situations where two of the answers are 0 and the other is 6. Thus, we must exclude those 3 examples (C(3,2)), making a total of 25 such situations.
solution = 25/216.

Approach Solution 2:

In essence, there is only one potential value for the third die if the numbers on the first two dice are known. So, you must only identify two-dice combinations that result in sums of 6 or higher. e.g. If the first die has a value of 1, the second die might produce either a 5 or a 6. The same is true for the first die, which has 3 possibilities for a result of 2, including 4 and 5. Yet, there are only 5 potential outcomes if the first die yields a number of 6.

The total number of outcomes comes out to be (2 + 3 + 4 + 5 + 6 + 5) = 25.

Probability is 25/216!

Approach Solution 3:

Sum of 12 can be achieved in following ways
6,5,1---Total cases = 3! = 6
6,4,2---Total cases = 3!= 6
6,3,3---Total cases = 3!/2! = 3
5,5,2---Total cases = 3!/2! = 3
5,4,3---Total cases = 3! = 6
4,4,4---Total cases = 3!/3! = 1
Total cases = 25
Probability = 25 * (1/6 * 1/6 * 1/6) = 25/216

“What is the probability of getting a sum of 12 when rolling 3 dice” - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been borrowed from the book “GMAT Official Guide Quantitative Review”.

To understand GMAT Problem Solving questions, applicants must possess fundamental qualitative skills. Quant tests a candidate's aptitude in reasoning and mathematics. The GMAT Quantitative test's problem-solving phase consists of a question and a list of possible responses. By using mathematics to answer the question, the candidate must select the appropriate response. The problem-solving section of the GMAT Quant topic is made up of very complicated math problems that must be solved by using the right math facts.

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