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# Out of seven models, all of different heights, five models

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CEO
Joined: 15 Aug 2003
Posts: 3467
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Kudos [?]: 718 [0], given: 781

Out of seven models, all of different heights, five models [#permalink]  23 Sep 2003, 13:55
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Out of seven models, all of different heights, five models will be chosen to pose for a photograph. If the five models are to stand in a line from shortest to tallest, and the fourth-tallest and sixth-tallest models cannot be adjacent, how many different arrangements of five models are possible?
Intern
Joined: 03 May 2003
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Ways to pick 5 models from 7
7C5 = 21

Ways in which 4th and 6th are adjacent = 4

21-4 = 17 ways
Intern
Joined: 16 Jul 2003
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I also got 17. But my approach was different.

Case I - ways when neither 4 nor 6 is selected = 5c5.
Case II - ways when either of the 2 ( #4or #6) is selected = 2c1*5c4.
Case III - ways when 4 and 6 are selected (here #5 is a must as 4 & 6
can't be adjacent and models are to arranged shortest to tallest. = 4c2 ( as #4,5,6 are to be selected )

Total ways, when 4 & 6 can't be adjacent and group is arranged from shortest to tallest = I+II+III = 1 + 10 + 6 = 17.

But

Can you pls. explain how did you get

Quote:
Ways in which 4th and 6th are adjacent = 4

Senior Manager
Joined: 21 Aug 2003
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First of all we are looking for arrangements not combinations/ways.

Total arrangements of 5 models = 7C5*5!------- a
But, the above arrangements include where 4th and 6the are together.
Number of ways when 4th & 6th are together = 5C3*4!------b
(Select 3 members from remaining 5 - 4th & 6th are already in group - and ways they can be arranged so that 4th & 6th remain togther = 4!)
Thus ways to arrange 5 models out of 7 such that 4 & 6 are not adjacent
= eqa - eqb
= 2280
Intern
Joined: 21 Jul 2003
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Location: India
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Vicky,
How does your solution account for the arrangement of models from shortest to tallest ? I guess there is something we are missing here
CEO
Joined: 15 Aug 2003
Posts: 3467
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Kudos [?]: 718 [0], given: 781

Exy has the best possible solution.

Vicks, there is no arrangement issue here..we know we have to arrrange from highest to lowest..
For every combo, we have only ONE way it will work

Thanks
praetorian
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Explination [#permalink]  24 Sep 2003, 15:08
The 4 ways in which 4 & 6 are adjacent are:
12346
23467
13467
12346
Senior Manager
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oh man.. my brain is working too much....
thanks praet...
Intern
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Re: Explination [#permalink]  25 Sep 2003, 04:42
exy18 wrote:
The 4 ways in which 4 & 6 are adjacent are:
12346
23467
13467
12346

Your first and last orders are the same. What's the 4th way?
Intern
Joined: 16 Jul 2003
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Quote:
Your first and last orders are the same. What's the 4th way?

Exy mistyped the 4th, it should be 12467.

His method is much better & faster than My method !
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