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
[spoiler]OA C[/spoiler]
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goes into a wrong envelope?
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sanju09
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GMATGuruNY
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Let the correct ordering be ABCD.sanju09 wrote: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
[spoiler]OA C[/spoiler]
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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.
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GMATGuruNY
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Formula approach:
A DERANGEMENT is a permutation in which NO element is in the correct position.
Given n elements (where n>1):
Number of possible derangements = n! (1/2! - 1/3! + 1/4! + ... + ((-1)^n)/n!)
Thus, given 4 letters:
Number of possible derangements = 4! (1/2! - 1/3! + 1/4!) = 12-4+1 = 9.
A DERANGEMENT is a permutation in which NO element is in the correct position.
Given n elements (where n>1):
Number of possible derangements = n! (1/2! - 1/3! + 1/4! + ... + ((-1)^n)/n!)
Thus, given 4 letters:
Number of possible derangements = 4! (1/2! - 1/3! + 1/4!) = 12-4+1 = 9.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
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My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
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I unlock the best way for YOU to solve problems.
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Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
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