Permutations Aah....

[coolhabhi]

Four letters are addressed to four different persons and the corresponding envelopes are prepared. The letters are put into the envelopes at random. What is the possible number of ways so that no letter is in its proper envelope?
A) 9
B) 12
C) 16
D) 17
E) 24

official answer - A

[Brent@GMATPrepNow]

Four letters are addressed to four different persons and the corresponding envelopes are prepared. The letters are put into the envelopes at random. What is the probability that no letter is in its proper envelope?
A) 9
B) 16
C) 24
D) 12

official answer - A

Are you sure you transcribed the question correctly?
Probabilities range from 0 to 1, so the 4 answer choices don't make any sense.

Cheers.
Brent

[[email protected]]

Hi coolhabhi,

What is the source of this question?

I ask because there are only 4 answer choices (not the normal 5 that you'll see on the GMAT) and the question doesn't match the answers (none of those answers, in their current format, can be the probability of anything).

GMAT assassins aren't born, they're made,
Rich

[GMATGuruNY]

I believe the following reflects the intent of the problem:

There are four letters A, B, C, and D that have to go into 4 envelopes addressed to a, b, c, and d respectively. In how many ways can the four letters be put in the 4 envelopes such that every letter goes into a wrong envelope?
A. 20
B. 12
C. 9
D. 6
E. 4

Let the correct ordering be ABCD.
Strategy:
Write out the possible arrangements for ONE CASE.
Use this information to determine the number of possible arrangements for the REMAINING CASES.

Case 1: A in the second position
The following arrangements are viable:
BADC
CADB
DABC
Total options = 3.

Implication:
When A is in the 3rd position, there will be 3 more options.
When A is in the 4th position, there will be 3 more options.

Total options = 3+3+3 = 9.

The correct answer is C.

[Brent@GMATPrepNow]

As Mitch has shown, this question does not require us to apply any complicated formulas/techniques. Instead, we can just list and count the number of possible outcomes.
How do we know that "listing and counting" may be a viable approach? Check the answer choices (NOTE: Always check the answer choices before you begin any kind of calculations). In this case, the answer choices are all pretty small, which means "listing and counting" may work nicely.

I write more about this in the following article:
https://www.gmatprepnow.com/articles/gma ... i-counting

As you might imagine, listing and counting won't be the best (i.e., fastest) strategy for every counting question. What's important, however, is that you understand its potential value and consider it a worthy candidate when considering your options.

Cheers,
Brent