A Password on Mr. Wallace's Briefcase Consists of 5 digits GMAT Problem Solving

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.

Suggested GMAT Problem Solving Samples

• If P2−QR=10P2−QR=10 ,Q2+PR=10Q2+PR=10 ,R2+PQ=10R2+PQ=10 GMAT Problem Solving
• If y (u-c) = 0 and j (u-k) = 0, Which of the Following Must be True, Assuming c < kc < k? GMAT Problem Solving
• What is the Remainder when 333^222 is Divided by 7? GMAT Problem Solving
• In a College of 300 Students, Every Student Reads 5 Newspapers and every Newspaper is Read by 60 Students GMAT Problem Solving
• If 4 People are Selected from a Group of 6 Married Couples, What is the Probability That none of Them would be Married to Each Other? GMAT Problem Solving
• If the Equation |x|+|y|= 5 Encloses a Certain Region on the Graph, What is the Area of that Region? GMAT Problem Solving
• If x = ¾ and y = ⅖ , What is the Value of √(x2+6x+9)(x2+6x+9) - √(y2−2y+1)(y2−2y+1)? GMAT Problem Solving
• A Chord of a Circle is Equal to its Radius. GMAT Problem Solving
• A Clock loses a Minute Every Three Hours for 4 Days and Gains 1% in the Subsequent 6 Days. GMAT Problem Solving
• The Population of the Bacteria Colony Doubles Every Day GMAT Problem Solving
• If tu=xytu=xyand ty=uxty=ux Where t, u, x, and y are Non-Zero Integers GMAT Problem Solving
• A Farm has Chickens, Cows and Sheep. The Number of Chickens and Cows Combined is 3 Times the Number of Sheep. GMAT Problem Solving
• In how Many Different Ways Can a Group of 8 People be Divided into 4 Teams of 2 People Each? GMAT Problem Solving
• If m is Three Times n, and if 2n + 3 is 20% of 25, What is the value of m? GMAT Problem Solving
• If Ben Were to Lose the Championship, Mike would be the Winner GMAT Problem Solving
• A Train Travelling at a Certain Constant Speed takes 30 seconds GMAT Problem Solving
• The product of the first 10 prime numbers is closest to which of the following? GMAT Problem Solving
• There is a 120 liter mixture of alcohol and water. The ratio of alcohol to water is 7 : 5 GMAT Problem Solving
• An equilateral triangle ABC is inscribed in square ADEF, forming three right triangles GMAT Problem Solving
• A Furniture Store Sells Only Two Models of Desks, Model A and Model B. The Selling Price of Model A is $120 GMAT Problem Solving
• A contractor combined x tons of a gravel mixture that contained 10 percent gravel G GMAT Problem Solving
• There are 100 Apples in a Bag of which 98% are Green and Rest are Red GMAT Problem Solving
• How many litres of a 90% solution of concentrated acid needs to be mixed with a 75% solution GMAT Problem Solving
• Jug Contains Water And Orange Juice In The Ratio 5:7 . Another Jug Contains Water And Orange J GMAT Problem Solving
• The number of ways in which 8 different flowers can be seated to form a garland so that 4 particular flowers are never separated GMAT Problem Solving
• A train travels from Albany to Syracuse, a distance of 120 miles, at an average rate of 50 miles per hour GMAT Problem Solving
• The hexagon ABCDEF is regular. That means all its sides are the same length and all its interior angles are the same size. GMAT Problem Solving
• y varies directly as x and when x = 6, y = 24. What is the value of y, when x = 5? GMAT Problem Solving
• If k is an Integer and 2 < k < 7, for How Many Different Values of k is There a Triangle With Sides of Lengths 2, 7, and k? GMAT Problem Solving
• How many factors does 36^2 have? GMAT Problem Solving
• A number when divided successively by 4 and 5 leaves remainder 1 and 4 respectively GMAT Problem Solving
• Walking at 6/7 th of his usual speed, a man is 25 min too late GMAT Problem Solving
• An Inlet Pipe can Fill in an Empty Cistern in 30 minutes Whereas a leak in the Bottom of the Cistern can Empty a Filled Tank in 40 minutes GMAT Problem Solving
• A milkman cheats his customers by adding water to the milk he sells GMAT Problem Solving
• if 80 lamps can be lighted, 5 hours per day for 10 days for $21.25, then the number of lamps, GMAT Problem Solving
• Few of the corporate contributions to the earthquake relief fund, aside from Pterocom GMAT Problem Solving
• X is Older Than Y, Z Is Younger Than W And V Is Older Than Y GMAT Problem Solving
• What is the Largest Power of 3 Contained in 200! GMAT Problem Solving
• Find the Greatest Number That Will Divide 43,91 and 183 So as to Leave GMAT Problem Solving
• A Right Angled Triangle has its Sides in Arithmetic Progression and Being Integers GMAT Problem Solving
• Out of 7 Consonants and 4 Vowels, How Many Words of 3 Consonants and 2 Vowels Can be Formed? GMAT Problem Solving
• 4 Bells Toll Together at 9:00 A.M. They Toll After 7, 8, 11 and 12 seconds Respectively GMAT Problem Solving
• A Man can Hit a Target Once with 4 Shots. If He Fires 4 Shots in Success GMAT Problem Solving
• A is twice as good a workman as B and together they finish a piece of GMAT Problem Solving
• Frances can complete a job in 12 hours, and Joan can complete the same GMAT Problem Solving
• The Average Age of Chief Executive Officers (CEO’s) in a Large Sample of Companies is 57 GMAT Problem Solving
• Running at the Same Constant Rate, 6 Identical Machines can GMAT Problem Solving
• If a and b are positive integers such that a – b and a/b are both even GMAT Problem Solving
• If g is an integer what is the value of(−1)g4−1(−1)g4−1? GMAT Problem Solving
• What is the Area of the Triangle with the following Vertices L(1,3) M(5,1) and N(3,5)? GMAT Problem Solving