Tanya prepared 4 different letters to be sent to 4 different addresses. For each letter, she prepared an envelope with its correct address. If the 4 letters are to be put into the 4 envelopes at random, what is the probability that only 1 letter will be put into the envelope with its correct address?
(A) 1/24
(B) 1/8
(C) 1/4
(D) 1/3
(E) 3/8
A DERANGEMENT is a permutation in which NO element is in the correct position.
Given n elements (where n>1):
Number of derangements = n! (1/2! - 1/3! + 1/4! + ... + ((-1)^n)/n!)
Let the 4 letters be ABCD.
Total arrangements:
Total number of ways to arrange the 4 letters = 4! = 24.
Good arrangements:
In a good arrangement, EXACTLY ONE letter is in the correct position.
Number of options for the one letter put into the correct position = 4. (A, B, C, or D)
Number of ways to DERANGE the 3 remaining letters = 3! (1/2! - 1/3!) = 3-1 = 2.
To combine these options, we multiply:
4*2 = 8.
Good/total = 8/24 = 1/3.
The correct answer is
D.
The following solution does not require knowledge of the derangement formula:
Let the correct order for the letters be A-B-C-D.
Case 1: Only A in the correct envelope
P(A is placed in the CORRECT envelope) = 1/4. (Of the 4 envelopes, 1 is correct.)
P(B is placed in an INCORRECT envelope) = 2/3. (Of the 3 remaining envelopes, only 1 is correct, so 2 are incorrect.)
At this point, B has taken either the envelope for C or the envelope for D.
Thus, of C and D, ONLY ONE could now be placed in the correct envelope.
P(this letter is placed in an INCORRECT envelope) = 1/2. (Of the 2 remaining envelopes, only 1 is correct, so 1 is incorrect.)
P(last letter is placed in an INCORRECT envelope) = 1. (As noted above, only ONE of C and D could be placed in the correct envelope, so the other must be placed in an incorrect envelope.)
Since all of these events must happen in order for A to be the only envelope correctly placed, we multiply the fractions:
1/4 * 2/3 * 1/2 * 1.
Other cases:
Since there are 4 options for the one envelope correctly placed -- A, B, C, or D -- the result above must be multiplied by 4:
1/4 * 2/3 * 1/2 * 1 * 4 = 1/3.
Last edited by
GMATGuruNY on Thu Nov 21, 2013 4:05 am, edited 2 times in total.
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