Bunuel wrote:
Let S be the set of permutations of the sequence 2, 3, 4, 5, 6 for which the first term is not 2. A permutation is chosen randomly from S. The probability that the second term is 5 is given by a/b (in lowest terms). What is a+b?
(A) 5
(B) 6
(C) 11
(D) 16
(E) 19
VERITAS PREP OFFICIAL SOLUTION:Most Common Solution:What are the permutations of sequence S? They are the different ways in which we can arrange the elements of S. For example, 3, 2, 4, 5, 6 or 4, 2, 3, 6, 5 or 6, 3, 4, 5, 2 etc
In how many different ways can we make the sequence? The first element can be chosen in 4 ways – one of 3, 4, 5 and 6. (You are given that 2 cannot be the first element).
The second element can be chosen in 4 ways (2 and the leftover 3 numbers).
The third element can be chosen in 3 ways.
The fourth element can be chosen in 2 ways.
And finally there will be only 1 element left for the last spot.
Number of ways of making set S = 4*4*3*2*1 = 96
In how many of these sets will 5 be in the second spot?
If 5 is reserved for the second spot, there are only 3 ways of filling the first spot (3 or 4 or 6).
The second spot has to be taken by 5.
The third element will be chosen in 3 ways (ignoring 5 and the first spot)
The fourth element can be chosen in 2 ways.
And finally there will be only 1 element left for the last spot.
Number of favorable cases = 3*1*3*2*1 = 18
Required Probability = Favorable Cases/Total Cases = 18/96 = 3/16 = a/b
a+b = 3 + 16 = 19
Answer (E)
Intellectual Approach:Use a bit of logic of symmetry to solve this question without any calculations.
Set S would include all such sequences as 3, 2, 4, 5, 6 or 4, 2, 3, 6, 5 or 6, 3, 4, 5, 2 etc – starting with 3, with 4, with 5 or with 6 with equal probability.
By symmetry, note that 1/4th of them will start with 5 – which we need to ignore – so we are left with the rest of the 3/4th sequences.
Now, in these 3/4th sequences which start with either 3 or 4 or 6, 5 could occupy any one of the 4 positions – second, third, fourth or fifth with equal probability. So we need 1/4th of these sequences i.e. only those sequences in which 5 is in the second spot.
Probability that 5 is the second element of the sequence = (3/4)*(1/4) = 3/16
Therefore, a+b = 3+16 = 19
Answer (E)
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