Ida had 5 Cards with Matching Envelopes- Same in Design, Different in GMAT Problem Solving

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.

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