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# Nine identical chips are numbered from 1 to 9 (one different number pe

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GMATH Teacher
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Joined: 12 Oct 2010
Posts: 935
Nine identical chips are numbered from 1 to 9 (one different number pe  [#permalink]

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25 Feb 2019, 08:06
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Difficulty:

65% (hard)

Question Stats:

44% (02:00) correct 56% (02:28) wrong based on 34 sessions

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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

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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
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Re: Nine identical chips are numbered from 1 to 9 (one different number pe  [#permalink]

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25 Feb 2019, 08:34
Top Contributor
fskilnik wrote:
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

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)

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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Re: Nine identical chips are numbered from 1 to 9 (one different number pe  [#permalink]

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25 Feb 2019, 13:52
fskilnik wrote:
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

$$?\,\,\mathop = \limits^{\left( * \right)} \,\,\,\# \,\,\left( {{\rm{odd,even,odd,even,odd,even,odd,even,odd}}} \right)\,\,{\rm{tuples}}$$

$$\left( * \right)\,\,{\rm{must}}\,\,{\rm{start}}\,\,{\rm{and}}\,\,{\rm{finish}}\,\,{\rm{with}}\,\,{\rm{odd \,\, numbers}}\,\,\,\left( {5\,\,{\rm{odd}}\,\,{\rm{numbers}}\,,\,4\,\,{\rm{even}}\,{\rm{numbers}}} \right)$$

$$?\,\, = \,\,{P_5} \cdot {P_4} = 5!\,\, \cdot 4!\,\, = \,\underleftrightarrow {\,120 \cdot 24 = 2 \cdot {{12}^2} \cdot 10} = 2880$$

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: Nine identical chips are numbered from 1 to 9 (one different number pe  [#permalink]

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27 Feb 2019, 04:48
fskilnik wrote:
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

odd = 1,3,5,7,9 ; 5 digits
and even = 2,4,6,8
so first odd then even can be taken out in sequence as;
5*4*4*3*3*2*2*1*1 = 2880
IMO D
Re: Nine identical chips are numbered from 1 to 9 (one different number pe   [#permalink] 27 Feb 2019, 04:48
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