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# Eight women of eight different heights are to pose for a pho

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Intern
Joined: 11 Nov 2009
Posts: 1
Eight women of eight different heights are to pose for a pho  [#permalink]

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04 Apr 2010, 20:53
4
13
00:00

Difficulty:

95% (hard)

Question Stats:

26% (02:13) correct 74% (02:11) wrong based on 241 sessions

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Eight women of eight different heights are to pose for a photo in two rows of four. Each woman in the second row must stand directly behind a shorter woman in the first row. In addition, all of the women in each row must be arranged in order of increasing height from left to right. Assuming that these restrictions are fully adhered to, in how many different ways can the women pose?

(A) 2
(B) 14
(C) 15
(D) 16
(E) 18
Math Expert
Joined: 02 Sep 2009
Posts: 52906
Re: Arrangement of Eight women  [#permalink]

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05 Apr 2010, 15:57
8
3
ajk wrote:
Eight women of eight different heights are to pose for a photo in two rows of four. Each woman in the second row must stand directly behind a shorter woman in the first row. In addition, all of the women in each row must be arranged in order of increasing height from left to right. Assuming that these restrictions are fully adhered to, in how many different ways can the women pose?

(A) 2
(B) 14
(C) 15
(D) 16
(E) 18

Shortest (1) and highest (8) women are fixed:
XXX8
1XXX

2 and 7 have two options, so total 4 patterns:

1:
2xx8
1xx7

|2468| |2568|
|1357| |1347|

2.
2x78
1xxx

|2478| |2578| |2678|
|1356| |1346| |1345|

3.
xxx8
12x7

|4568| |3468| |3568|
|1237| |1257| |1247|

4.
xx78
12xx

|3478| |3678| |3578| |5678| |4578| |4678|
|1256| |1245| |1246| |1234| |1236| |1235|

14 options.

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Re: Arrangement of Eight women  [#permalink]

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04 Apr 2010, 21:05
I doubt this is a Gmat question..what is the source?
we will have to actually count the cases here.
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Re: Arrangement of Eight women  [#permalink]

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05 Apr 2010, 07:48
Is it just me or is there only 1 way to arrange the women given the criteria.

The criteria basically boil down to: arrange in ascending order in terms of height. Since they are all of different heights, this is only possible in 1 way...
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Joined: 10 Aug 2009
Posts: 116
Re: Arrangement of Eight women  [#permalink]

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05 Apr 2010, 07:50
No, nevermind, I'm wrong. In any case I have no idea how to solve this problem.
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Joined: 15 Dec 2009
Posts: 279
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Re: Arrangement of Eight women  [#permalink]

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05 Apr 2010, 10:43
yeah this is not gmat, this is a problem where you really have to count it out. I have come up with 8+ so far, but I'm giving up
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Re: Arrangement of Eight women  [#permalink]

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05 Apr 2010, 15:36
i would have to go with D 16. you don't have to count it out fully, but it would take a long time. i found a pattern by placing the women in certain areas. for example: if you label each woman numbers 1 - 8, then you know the lowest and highest numbers can only occupy one area so that gives us a total of 2.

the second woman (2) can only occupy two spaces since she is larger than 1, but none other. she can either stand next to woman one or behind her.Total: 2

woman number 3 can occupy 3 spaces. two on the bottom and one on the top because she is larger than 1 and 2 and must either stand beside them or above them. there is an interesting pattern if you draw it out.
5 6 7 8
1 2 3 4
OR
2 6 7 8
1 3 4 5
OR
3 6 7 8
1 2 4 5
notice the color and place of 3 and how it shifts. Total 3.

now woman 4. Total 3
5 6 7 8
1 2 3 4
OR
3 5 7 8
1 2 4 6
OR
4 5 7 8
1 2 3 6
notice how the numbers shift once to the right and are diagonal on the top. in fact numbers 2 - 6 seem to share this property. 7 follows the properties of woman 2 and therefore has two spaces. if we write this out we have: #1 has 1, #2 has 2, #3 has 3, #4 has #3, #5 has 3, #6 has 3, #7 has 2, #8 has 1. this gives us a total of 18 ways, but i think there are two ways that are at least double counted and need to be subtracted. 18 - 2 = 16. hopefully i explained this well. it was pretty tough and took me about 6 or 7 minutes. WAY TOO LONG. anyone else get 16? or maybe a different answer?
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Re: Arrangement of Eight women  [#permalink]

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05 Apr 2010, 15:38
By the way, whats the OA?
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Re: Arrangement of Eight women  [#permalink]

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10 Apr 2010, 03:08
This was a question for Bunuel only :D
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Joined: 05 Mar 2010
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Re: Arrangement of Eight women  [#permalink]

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10 Apr 2010, 08:24
yup for bunuel only and 800+ question as per GMAT standard
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Re: Arrangement of Eight women  [#permalink]

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29 Dec 2013, 16:53
Bunuel wrote:
ajk wrote:
Eight women of eight different heights are to pose for a photo in two rows of four. Each woman in the second row must stand directly behind a shorter woman in the first row. In addition, all of the women in each row must be arranged in order of increasing height from left to right. Assuming that these restrictions are fully adhered to, in how many different ways can the women pose?

(A) 2
(B) 14
(C) 15
(D) 16
(E) 18

Shortest (1) and highest (8) women are fixed:
XXX8
1XXX

2 and 7 have two options, so total 4 patterns:

1:
2xx8
1xx7

|2468| |2568|
|1357| |1347|

2.
2x78
1xxx

|2478| |2578| |2678|
|1356| |1346| |1345|

3.
xxx8
12x7

|4568| |3468| |3568|
|1237| |1257| |1247|

4.
xx78
12xx

|3478| |3678| |3578| |5678| |4578| |4678|
|1256| |1245| |1246| |1234| |1236| |1235|

14 options.

Bunuel, should one study this question for GMAT prep?

Cheers!
J
Math Expert
Joined: 02 Sep 2009
Posts: 52906
Re: Arrangement of Eight women  [#permalink]

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30 Dec 2013, 00:58
jlgdr wrote:
Bunuel wrote:
ajk wrote:
Eight women of eight different heights are to pose for a photo in two rows of four. Each woman in the second row must stand directly behind a shorter woman in the first row. In addition, all of the women in each row must be arranged in order of increasing height from left to right. Assuming that these restrictions are fully adhered to, in how many different ways can the women pose?

(A) 2
(B) 14
(C) 15
(D) 16
(E) 18

Shortest (1) and highest (8) women are fixed:
XXX8
1XXX

2 and 7 have two options, so total 4 patterns:

1:
2xx8
1xx7

|2468| |2568|
|1357| |1347|

2.
2x78
1xxx

|2478| |2578| |2678|
|1356| |1346| |1345|

3.
xxx8
12x7

|4568| |3468| |3568|
|1237| |1257| |1247|

4.
xx78
12xx

|3478| |3678| |3578| |5678| |4578| |4678|
|1256| |1245| |1246| |1234| |1236| |1235|

14 options.

Bunuel, should one study this question for GMAT prep?

Cheers!
J

I think that this question is beyond the scope.
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Re: Eight women of eight different heights are to pose for a pho  [#permalink]

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31 Jan 2015, 08:24
2
1
ajk wrote:
Eight women of eight different heights are to pose for a photo in two rows of four. Each woman in the second row must stand directly behind a shorter woman in the first row. In addition, all of the women in each row must be arranged in order of increasing height from left to right. Assuming that these restrictions are fully adhered to, in how many different ways can the women pose?

(A) 2
(B) 14
(C) 15
(D) 16
(E) 18

i found this answer in one of the solution books:

This is a counting problem that is best solved using logic. First, let’s represent the line of women as follows:

0000
0000

where the heights go from 1 to 8 in increasing order and the unknowns are designated 0s. Since the women are arranged by their heights in increasing order from left to right and front to back, we know that at a minimum, the lineup must confirm to this:

0008
1000

Let’s further designate the arrangement by labeling the other individuals in the top row as X, Y and Z, and the individuals in the bottom row as A, B, and C.

XYZ8
1ABC

Note that Z must be greater than at least 5 numbers (X, Y, B, A, and 1) and less than at least 1 number (8). This means that Z can only be a 6 or a 7.

Note that Y must be greater than at least 3 numbers (X, A and 1) and less than at least 2 numbers (8 and Z). This means that Y can only be 4, 5, or 6.

Note that X must be greater than at least 1 number (1) and less than at least 3 numbers (8, Z and Y), This means that X must be 2, 3, 4, or 5.
This is enough information to start counting the total number of possibilities for the top row.

It will be easiest to use the middle unknown value Y as our starting point. As we determined above, Y can only be 4, 5, or 6.

Let’s check each case, making our conclusions logically:

If Y is 4, Z has 2 options (6 or 7) and X has 2 options (2 or 3). This yields 2 x 2 = 4 possibilities.

If Y is 5, Z has 2 options (6 or 7), and X has 3 options (2, 3, or 4). This yields 2 x 3 = 6 possibilities.

If Y is 6, Z has 1 option (7), and X has 4 options (2, 3, 4, or 5). This yields 1 x 4 = 4 possibilities.

For each of the possibilities above, the bottom row is completely determined because we have 3 numbers left, all of which must be in placed increasing order.
Hence, there are 4 + 6 + 4 = 14 ways for the women to pose.
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Joined: 20 Feb 2016
Posts: 4
Eight women of eight different heights are to pose for a pho  [#permalink]

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20 Aug 2016, 06:57
Another quick way to solve the question is:

Women can be rranged in ascending order as:
W1<W2<W3<W4<W5<W6<W7<W8

And the two rows:
- - - - R1
- - - - R2

W1 can be placed in two different ways (Either R1 or R2)
And as per the hierarchy you can place any of the 7 other women behind W1 (as they would all be more than W1 in height)

Probability - 7*2 - 14 ways
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Joined: 04 Apr 2015
Posts: 40
Re: Eight women of eight different heights are to pose for a pho  [#permalink]

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12 Oct 2016, 04:28
akankshasbisen wrote:
Another quick way to solve the question is:

Women can be rranged in ascending order as:
W1<W2<W3<W4<W5<W6<W7<W8

And the two rows:
- - - - R1
- - - - R2

W1 can be placed in two different ways (Either R1 or R2)
And as per the hierarchy you can place any of the 7 other women behind W1 (as they would all be more than W1 in height)

Probability - 7*2 - 14 ways

How W1 can be placed in 2 different ways. If you plae W1 in 2nd row any woman standing in front of W1 will be taller than W1. I think position of W1 is fixed.
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Posts: 624
Location: India
Re: Eight women of eight different heights are to pose for a pho  [#permalink]

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02 Dec 2016, 22:50
1
Top Contributor
ajk wrote:
Eight women of eight different heights are to pose for a photo in two rows of four. Each woman in the second row must stand directly behind a shorter woman in the first row. In addition, all of the women in each row must be arranged in order of increasing height from left to right. Assuming that these restrictions are fully adhered to, in how many different ways can the women pose?

(A) 2
(B) 14
(C) 15
(D) 16
(E) 18

This is how it looks with the position of the shortest and tallest fixed

- - - T
S - - -

The remaining are 234567

Keep selecting the leftmost 3 for the bottom row Place the reamaining 3 at the top row.

For example we would start with S234 , S235 , s236, s237 and then s245 and so on for the bottom rows. There would be only 14 valid options.
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Re: Eight women of eight different heights are to pose for a pho  [#permalink]

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23 Jun 2017, 17:48
1
it should not be hard after all, yup I just come up with the answer out of luck.
The solution is 4C1 + 4C2 + 4C3
The heights are followed:
5 6 7 8
1 2 3 4

Certainly, ones can exchange the number 4 with any number in the top row. Next, both number 3 and 4 with 2 numbers in the top row. Lastly, 2,3,4 with 3 numbers in top row.
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Re: Eight women of eight different heights are to pose for a pho  [#permalink]

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Re: Eight women of eight different heights are to pose for a pho   [#permalink] 08 Sep 2018, 11:00
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