Question: A password on Mr. Wallace's briefcase consists of 5 digits. What is the probability that the password contains exactly three digits 6?

1. 860 /90,000
2. 810/100,000
3. 858/100,000
4. 860/100,000
5. 1530/100,000

Correct Answer: A
Solution and Explanation:
Approach Solution 1:

How many possibilities are there in which the digit 6 fills three positions and 1 and 2 fill the remaining two positions?
Let's examine the results of the calculation 9*8*5!/(3!) = 1440 ways.
-> 9*8 suggests that we are taking into account all possible arrangements involving two digits. Since we used the numbers 1 and 2 as our examples, we will think about the configurations 1 2 and 2 1 individually.

Let's now investigate the sequence 1 2 followed by 3 6s. This arrangement of 1 2 6 6 6 is multiplied by 5!/(3! ), taking into account circumstances when 2 comes before 1. For instance, 2 1 6 6 6 is sometimes regarded as 5!/(3!) since it contains all conceivable combinations.
The arrangement 2 1 6 6 6 is consequently multiplied with 5!/(3!) independently, leading to the double counting since all arrangements pertaining to three 6's and 1,2 are already tallied because, as was previously mentioned, 9*8 views the arrangement 2 1 as distinct from 1 2.

In order to avoid duplicate counting, you should divide the calculation 9*8*5!/(3!) by 2!
The solution is 720 + 90 (the number of ways that two other digits are the same, i.e., 9*1*5!/(3!*2!)), which equals 810).

B is the correct answer.

Approach Solution 2:

We'll look at two situations:

Case I: We have two DIFFERENT numbers and three 6s (e.g., 66612)
Case II: We have two IDENTICAL digits and three 6s (e.g., 66677)

Case I: We have two DIFFERENT numbers and three 6s (e.g., 66612)
There are already three sixes. so that we can choose two distinct digits from (0,1,2,3,4,5,7,8 and 9)
We have 36 options available to us, or 9C2 ways.
After choosing our five digits, we must arrange them, which requires us to apply the Mississippi rule.
In 5!/3! ways, which equals 20, we can arrange 3 identical numerals and 2 distinct digits.
The number of ways to have three 6s and two DIFFERENT numbers is (36)(20)/(3+6), or 720.

Case II: We have two IDENTICAL digits and three 6s (e.g., 66677)
There are already three sixes. Therefore, we must choose one digit to replicate.
We can choose 1 digit from the range (0,1,2,3,4,5,7,8,9) in 9 different ways.
After choosing our five digits, we must arrange them, which requires us to apply the Mississippi rule.
3 identical 6s and 2 other identical digits can be arranged in 5!/3!2! ways, which equals 10 ways.
The number of ways in which we can have three 6s and two IDENTICAL numerals is (9)(10), or 90.

720 + 90 = 810 is the total number of ways to have three 6s.

There are 100,000 different 5-digit codes that can be used, hence P(having precisely three 6s) = 810/100,000.

B is the correct answer.

“A password on Mr. Wallace's briefcase consists of 5 digits" - 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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