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10 Feb 2015, 09:26
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E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
75%
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Question Stats:
55% (02:35) correct
45%
(02:11)
wrong
based on 123
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Let Q be the set of permutations of the sequence 1,2,3,4,5 for which the first term is not 1. A permutation is chosen randomly from Q. The probability that the second term is 3, in lowest terms, is a/b. What is a+b?
A) 5
B) 6
C) 11
D) 16
E) 19
A) 5
B) 6
C) 11
D) 16
E) 19
10 Feb 2015, 10:46
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manpreetsingh86
Let Q be the set of permutations of the sequence 1,2,3,4,5 for which the first term is not 1. A permutation is chosen randomly from Q. The probability that the second term is 3, in lowest terms, is a/b. What is a+b?
A) 5
B) 6
C) 11
D) 16
E) 19
Dear manpreetsingh86,A) 5
B) 6
C) 11
D) 16
E) 19
Wow! This is a good juicy hard question! I like it!
For a discussion of probability and counting techniques, see:
https://magoosh.com/gmat/2013/gmat-prob ... echniques/
OK, so first, we figure out the size of Q. As always, start counting with the most restrictive case. Since 1 cannot be in the first term, there are four possibilities for the first term; then 4 left for the second term; then 3, then 2, then 1.
N = 4*4*3*2*1 = 16*6 = 96
That's the denominator of our big probability fraction, the total number of possible permutations in Q
Now, for the numerator, the second term is the most restrictive: we have to put 3 in the second term. That's only 1 option in that slot. That leaves only 3 possible choices for the first term, which can be neither 3 nor 1. Then, once the first two are selected, we have 3 choices for the third term, 2 for the second, and 1 for the fifth.
N = 1*3*3*2*1 = 18
Probability = 18/96 = 3/16
3 + 16 = 19. Choice (E).
That was a fun problem! If anyone has any questions, please let me know.
Mike
11 Feb 2015, 00:40
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manpreetsingh86
Let Q be the set of permutations of the sequence 1,2,3,4,5 for which the first term is not 1. A permutation is chosen randomly from Q. The probability that the second term is 3, in lowest terms, is a/b. What is a+b?
A) 5
B) 6
C) 11
D) 16
E) 19
A) 5
B) 6
C) 11
D) 16
E) 19
Use a bit of logic of symmetry to solve this question without any calculations.
Q would include all such sets {2, 3, 1, 4, 5}, {3, 1, 2, 4, 5}, (4, 3, 2, 5, 1), {5, 3, 2, 1, 4} etc - starting with 2, with 3, with 4 or with 5.
Note that 1/4th of them will start with 3 so we are left with the rest of the 3/4th sets.
Now, in these 3/4th sets which start with either 2 or 4 or 5, 3 could occupy any one of the 4 positions - second, third, fourth or fifth with equal probability. So we need 1/4th of these sets.
Probability that 3 is the second element of Q = (3/4)*(1/4) = 3/16
So a+b = 3+16 = 19
Answer (E)
