Question: Ida had 5 cards with matching envelopes- same in design, different in color. She removed the cards from all envelopes and randomly put them back. What is the probability that exactly one card got into the matching envelope?

A. \(\frac{3}{16}\)
B. \(\frac{5}{8}\)
C. \(\frac{3}{8}\)
D. \(\frac{1}{4}\)
E. \(\frac{1}{2}\)

Answer:
Approach Solution (1):
Total number of ways = 5! = 120
Total number of given cards = 5
So chances of selecting 1 correct card into 1 correct envelope = 5C1
Total ways of choosing a card for an envelope = 5!
Now we have to select ways to select a card which is to put into wrong envelope
So for 4 cards we have 3 ways to choose wrong envelope
For 3 cards we have 3 ways to choose wrong envelope
And for 2 cards we have 1 way to choose wrong envelope
Total ways we can put 4 cards in wrong envelope = 3 * 3 * 1 = 9
The probability that exactly one card got into the matching envelope =\(^5C_1*{9\over5}={3\over8}\)
Correct option: C

Approach Solution (2):
Selecting one right envelope = 5C1 = 5 ways
De-arrangements of N things can be found by
\(N!({1\over2!}-{1\over3!}+{1\over4!}-...)\)
With alternative negative signs.
De-arrangements of 4 envelopes =\(4!(1-{1\over1!}+{1\over2!}-{1\over3!}+{1\over4!})=9\)
Total ways of arrangements = 120
Probability of 1 envelope placed in right place =\(9*{5\over120}={3\over8}\)
Correct option: C

Approach Solution (3):
Ida has 5 cards with matching envelopes – same in design, different in color
Cards:\(C_1,C_2,C_3,C_4,C_5\)
Envelopes:\(E_1,E_2,E_3,E_4,E_5\)(same design, different color)
The probability that exactly one card got into the matching envelope
Probability = Desired results / Total number of results
5 cards can be put into 5! Ways = 120
Exact one card correct:
Number of ways one card can be chosen which go into the correct envelope out of 5 cards is = 5C1 = 5
Number of ways\(C_2\)goes into the wrong envelope:\(E_3,E_4,E_5\)= 3 ways
Number of ways\(C_3\)goes into the wrong envelope:\(E_2,E_4,E_5\)= 3 ways
Number of ways\(C_4\)goes into the wrong envelope:\(E_5\)= 1 way
Number of ways\(C_5\)goes into the wrong envelope:\(E_4\)= 1 way
Total ways: 3 * 3 * 1 * 1 = 9
Desired results = 5 * 9
Total results = 120
Probability =\(9*{5\over120}={45\over120}={3\over8}\)
Correct option: C

“Ida had 5 cards with matching envelopes- same in design, different in color. She removed the cards from all envelopes and randomly put them back. What is the probability that exactly one card got into the matching envelope?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Suggested GMAT Problem Solving Questions:

• Two Consultants, Mary and Jim, Can Type up a Report in 12.5 Hours and Edit it in 7.5 Hours. GMAT Problem Solving
• GMAT Problem Solving - The Price of Raw Materials Has Gone up by 15%
• Greg Assembles Units of a Certain Product at a Factory. GMAT Problem Solving
• After 6 Games, Team B Had an Average of 61.5 Points Per Game. GMAT Problem Solving
• If 10, 12 and ‘x’ are Sides of an Acute Angled Triangle, How Many Integer Values of ‘x’ are Possible? GMAT Problem Solving
• \(\frac{15^{11}-15^{10}}{14}=?\) GMAT Problem Solving
• For Any Four Digit Number, abcd, *abcd*= (3^a)(5^b)(7^c)(11^d) GMAT Problem Solving
• GMAT Problem Solving - Iqbal Dealt Some Cards to Mushtaq and himself from a Full Pack of Playing Cards
• A Rectangular Box has Dimensions 12*10*8 Inches GMAT Problem Solving
• √9+√80+√9−√80)2 GMAT Problem Solving
• What Is The Circumference Of The Semicircle In The Figure Shown? GMAT Problem Solving
• In a Trapezium ABCD, AB Is Parallel To DC, AB = 3*DC, And The Diagonals Of The Trapezium Intersect At O GMAT Problem Solving
• A Bag Contains 5 Red Balls And Some Blue Balls GMAT Problem Solving
• A Small Water Pump Would Take 2 Hours To Fill An Empty Tank GMAT Problem Solving
• ABCD Is An Isosceles Trapezium GMAT Problem Solving
• An Equilateral Triangle is Constructed on Each Side of a Square GMAT Problem Solving
• For A Negative Integer x, What Is The Value Of x * √(x^2)? GMAT Problem Solving
• Five Letters Are To Be Placed Into Five Addressed Envelopes GMAT Problem Solving
• Find The Number Of Trailing Zeros In The Product Of (1^1) GMAT Problem Solving
• Find The Altitude Of An Equilateral Triangle Whose Side is 20 GMAT Problem Solving
• Which of the following is the value of √3√0.000064? GMAT Problem Solving
• Two Dice are Thrown Simultaneously. What is the Probability of Getting Two Numbers Whose Product is Even? GMAT Problem Solving
• During a Trip, Francine Traveled x Percent of the Total Distance at an Average Speed of 40 Miles GMAT Problem Solving
• Which of the Following Expressions CANNOT have a Negative Value? GMAT Problem Solving
• Which of the following is greatest? GMAT Problem Solving
• The Cost Price of 20 Articles is The Same as The Selling Price of x Articles GMAT Problem Solving
• In The Figure Shown, If The Area of The Shaded Region is 3 Times The Area of The Smaller Circular Region GMAT Problem Solving
• A Regular Hexagon has a Perimeter of 30 units GMAT Problem Solving
• All the Numbers 2, 3, 4, 5, 6, 7 are Assigned to the Six Faces of a Cube, One Number to Each Face GMAT Problem Solving
• If it is true that x > -2 and x < 7, which of the following must be true? GMAT Problem Solving
• In a Drawer, There are 4 White Socks, 3 Blue Socks, and 5 Grey Socks GMAT Problem Solving
• There are 100 Apples in a Bag of which 98% are Green and Rest are Red GMAT Problem Solving
• An Inlet Pipe can Fill in an Empty Cistern in 30 minutes GMAT Problem Solving
• A Zookeeper Counted the Heads of the Animals in a Zoo and Found it to be 80 GMAT Problem Solving
• A Shop Stores x kg of Rice. The First Customer Buys half this Amount Plus half a kg of Rice GMAT Problem Solving
• In a Class of 120 Students Numbered 1 to 120, All Even Numbered Students Opt for Physics GMAT Problem Solving
• Machine A Produces bolts at a Uniform Rate of 120 Every 40 seconds 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 GMAT Problem Solving
• 12 Marbles are Selected at Random from a Large Collection of White, Red, Green and Yellow Marbles GMAT Problem Solving
• Find the greatest number that will divide 43, 91 and 183 GMAT Problem Solving
• Of the 150 Houses in a Certain Development GMAT Problem Solving
• A man can hit a target once in 4 shots. If he fires 4 shots in succession GMAT Problem Solving
• If 75 Percent of a Class Answered the First Question on a Certain Test Correctly GMAT Problem Solving
• A Clock Strikes 4 taking 9 seconds. GMAT Problem Solving
• What is the Largest Power of 3 Contained in 200! GMAT Problem Solving
• Find The Value Of x GMAT Problem Solving
• GMAT Problem Solving - Which of the following expressions has the greatest values?
• GMAT Problem Solving – If @ x=x^2/2x^2-2 , What is the Units Digit of @ (@4)?
• GMAT Problem Solving – What is the product of all possible solutions of the equation |x+2|- 5|x+2| = -6?