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25 Feb 2019, 08:06
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
(hard)
Question Stats:
56% (02:05) correct
44%
(02:01)
wrong
based on 129
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GMATH practice exercise (Quant Class 18)
Nine identical chips are numbered from 1 to 9 (one different number per chip) and placed in a box. There are N ways in which all the chips are taken out from the box, one at a time and without repositions, in a sequence of alternating odd and even numbers. The value of N is:
(A) less than 1400
(B) between 1400 and 2000
(C) between 2000 and 2600
(D) between 2600 and 3200
(E) greater than 3200
Nine identical chips are numbered from 1 to 9 (one different number per chip) and placed in a box. There are N ways in which all the chips are taken out from the box, one at a time and without repositions, in a sequence of alternating odd and even numbers. The value of N is:
(A) less than 1400
(B) between 1400 and 2000
(C) between 2000 and 2600
(D) between 2600 and 3200
(E) greater than 3200
25 Feb 2019, 08:34
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fskilnik
ODDS: 1, 3, 5, 7, 9
EVENS: 2, 4, 6, 8
Take the task of removing the 9 chips and break it into stages.
Stage 1: Select an ODD number to be the 1st selection
There are 5 ODDs to choose from.
So, we can complete stage 1 in 5 ways
Stage 2: Select an EVEN number to be the 2nd selection
There are 4 EVENs to choose from.
So, we can complete stage 2 in 4 ways
Stage 3: Select an ODD number to be the 3rd selection
There are 4 ODDs remaining. So, we can complete this stage in 4 ways
Stage 4: Select an EVEN number to be the 4th selection
There are 3 EVENs remaining. So, we can complete this stage in 3 ways
Stage 5: Select an ODD number to be the 5th selection
There are 3 ODDs remaining. So, we can complete this stage in 3 ways
Stage 6: Select an EVEN number to be the 6th selection
There are 2 EVENs remaining. So, we can complete this stage in 2 ways
Stage 7: Select an ODD number to be the 7th selection
There are 2 ODDs remaining. So, we can complete this stage in 2 ways
Stage 8: Select an EVEN number to be the 8th selection
There is 1 EVEN number remaining. So, we can complete this stage in 1 way
Stage 9: Select an ODD number to be the 9th selection
There is 1 ODD number remaining. So, we can complete this stage in 1 way
By the Fundamental Counting Principle (FCP), we can complete all 9 stages in (5)(4)(4)(3)(3)(2)(2)(1)(1) ways (= 2880 ways)
Answer: D
Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. So, be sure to learn it.
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25 Feb 2019, 13:52
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fskilnik
\(?\,\,\mathop = \limits^{\left( * \right)} \,\,\,\# \,\,\left( {{\rm{odd,even,odd,even,odd,even,odd,even,odd}}} \right)\,\,{\rm{tuples}}\)
\(\left( * \right)\,\,{\rm{must}}\,\,{\rm{start}}\,\,{\rm{and}}\,\,{\rm{finish}}\,\,{\rm{with}}\,\,{\rm{odd \,\, numbers}}\,\,\,\left( {5\,\,{\rm{odd}}\,\,{\rm{numbers}}\,,\,4\,\,{\rm{even}}\,{\rm{numbers}}} \right)\)
\(?\,\, = \,\,{P_5} \cdot {P_4} = 5!\,\, \cdot 4!\,\, = \,\underleftrightarrow {\,120 \cdot 24 = 2 \cdot {{12}^2} \cdot 10} = 2880\)
The correct answer is (D).
We follow the notations and rationale taught in the GMATH method.
Regards,
Fabio.
