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

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

by fskilnik@GMATH » Mon Feb 25, 2019 7:01 am

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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

Answer: [spoiler]____(D)__[/spoiler]
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by Brent@GMATPrepNow » Mon Feb 25, 2019 7:23 am
fskilnik@GMATH 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

Answer: [spoiler]____(D)__[/spoiler]

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. For more information about the FCP, watch this video: https://www.gmatprepnow.com/module/gmat- ... /video/775

You can also watch a demonstration of the FCP in action: https://www.gmatprepnow.com/module/gmat ... /video/776

Then you can try solving the following questions:

EASY
- https://www.beatthegmat.com/what-should- ... 67256.html
- https://www.beatthegmat.com/counting-pro ... 44302.html
- https://www.beatthegmat.com/picking-a-5- ... 73110.html
- https://www.beatthegmat.com/permutation- ... 57412.html
- https://www.beatthegmat.com/simple-one-t270061.html


MEDIUM
- https://www.beatthegmat.com/combinatoric ... 73194.html
- https://www.beatthegmat.com/arabian-hors ... 50703.html
- https://www.beatthegmat.com/sub-sets-pro ... 73337.html
- https://www.beatthegmat.com/combinatoric ... 73180.html
- https://www.beatthegmat.com/digits-numbers-t270127.html
- https://www.beatthegmat.com/doubt-on-sep ... 71047.html
- https://www.beatthegmat.com/combinatoric ... 67079.html


DIFFICULT
- https://www.beatthegmat.com/wonderful-p- ... 71001.html
- https://www.beatthegmat.com/permutation- ... 73915.html
- https://www.beatthegmat.com/permutation-t122873.html
- https://www.beatthegmat.com/no-two-ladie ... 75661.html
- https://www.beatthegmat.com/combinations-t123249.html


Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by fskilnik@GMATH » Mon Feb 25, 2019 12:39 pm
fskilnik@GMATH 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

$$?\,\,\mathop = \limits^{\left( * \right)} \,\,\,\# \,\,\left( {{\text{odd,even,odd,even,odd,even,odd,even,odd}}} \right)\,\,{\text{tuples}}$$
$$\left( * \right)\,\,{\text{must}}\,\,{\text{start}}\,\,{\text{and}}\,\,{\text{finish}}\,\,{\text{with}}\,\,{\text{odd numbers}}\,\,\,\left( {5\,\,{\text{odd}}\,\,{\text{numbers}}\,,\,4\,\,{\text{even}}\,{\text{numbers}}} \right)$$

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


The correct answer is (D).


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

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
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