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Goldenrod and No Hope are in a horse race with 6 contestants [#permalink]

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10 Aug 2009, 20:40

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Goldenrod and No Hope are in a horse race with 6 contestants. How many different arrangements of finishes are there if No Hope always finishes before Goldenrod and if all of the horses finish the race?

Goldenrod and No Hope are in a horse race with 6 contestants. How many different arrangements of finishes are there if No Hope always finishes before Goldenrod and if all of the horses finish the race?

(A) 720 (B) 360 (C) 120 (D) 24 (E) 21

All 6 horses can finish the race in 6! way (assuming no tie).

If no tie is possible between No Hope and Goldenrod, then in half of these cases No Hope will be before Goldenrod and in half of these cases after (not necessarily right before or right after). How else? So, there are 6!/2=360 different arrangements of finishes where No Hope always finishes before Goldenrod.

Re: Goldenrod and No Hope are in a horse race with 6 contestants [#permalink]

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23 May 2013, 07:48

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I have solved this in below fashion

Attachment:

gamtQ2.png [ 2.97 KiB | Viewed 6935 times ]

So, When 'N' finishes first then there are 5 places where G can take (i.e 5 ways) When 'N' finishes second then there are 4 places where G can take (i.e 4 ways) When 'N' finishes third then there are 3 places where G can take (i.e 3 ways) ...

Re: Goldenrod and No Hope are in a horse race with 6 contestants [#permalink]

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23 May 2013, 08:39

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vishnuvardhan777 wrote:

I have solved this in below fashion

When 'N' finishes second then there are 4 places where G can take (i.e 4 ways) When 'N' finishes third then there are 3 places where G can take (i.e 3 ways) ...

No of ways is 5*4*3*2 = 120.

please explain flaw in my explanation.

Thanks in Advance

The most effective way to solve this is Bunuel's. However if we wanna take your approach more calculus are needed:

first of all 6 spots to fill _ _ _ _ _ _ If 'N' finishes first then the other 5 spots can be filled in \(5!\) ways N 5 4 3 2 1

And now it gets complicated... If 'N' second then the other 5 spots can be filled in \(4*4!\) ways 4 N 4 3 2 1. Why this? N is in second position and there are 5 horses left. Of those only 4 can occupy the first position (every horse EXCEPT g), so write 4 on the first line. We have 4 slots left and 4 horses => 4 3 2 1 for the remaining spots

With the same method if N finished 3rd, we get \(4*3*3!\) ways 4 3 N 3 2 1 N on the third slot, 5 horses left. The first spot can be occupied by 4 horses (every horse EXCEPT g), the second can be occupied by three horses (every horse EXCEPT g and the previusly chosen one); then for the others spots we have 3 horses => 3! ways. if N finished 4th, we get \(4*3*2*2!\) ways 4 3 2 N 2 1 if N finished 5th, we get \(4*3*2*1\) ways 4 3 2 1 N 1

Sum them up and you get 360. This approach however is really long

Hope it's clear
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So, When 'N' finishes first then there are 5 places where G can take (i.e 5 ways) When 'N' finishes second then there are 4 places where G can take (i.e 4 ways) When 'N' finishes third then there are 3 places where G can take (i.e 3 ways) ...

No of ways is 5*4*3*2 = 120.

please explain flaw in my explanation.

Thanks in Advance

If N is on the first place G can take ANY of the remaining 5 places and the remaining 4 horses can be arranged in 4! number of ways: 5*4!=120; If N is on the second place G can take 4 places and the remaining 4 horses can be arranged in 4! number of ways: 4*4!=96; If N is on the third place G can take 3 places and the remaining 4 horses can be arranged in 4! number of ways: 3*4!=72; If N is on the fourth place G can take 2 places and the remaining 4 horses can be arranged in 4! number of ways: 2*4!=48; If N is on the fifth place G can take only 1 place and the remaining 4 horses can be arranged in 4! number of ways: 1*4!=24;

So, When 'N' finishes first then there are 5 places where G can take (i.e 5 ways) When 'N' finishes second then there are 4 places where G can take (i.e 4 ways) When 'N' finishes third then there are 3 places where G can take (i.e 3 ways) ...

Re: Goldenrod and No Hope are in a horse race with 6 contestants [#permalink]

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08 Sep 2014, 08:40

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: Goldenrod and No Hope are in a horse race with 6 contestants [#permalink]

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29 Nov 2014, 07:07

@Bunuel,I used the method you explained later but i want to use the first method(All cases by 2) next time..How do i come to know in which questions i can use this approach..that in half cases one thing must have happened and in the other half cases the other thing?

Re: Goldenrod and No Hope are in a horse race with 6 contestants [#permalink]

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29 Nov 2014, 09:57

Bunuel wrote:

Goldenrod and No Hope are in a horse race with 6 contestants. How many different arrangements of finishes are there if No Hope always finishes before Goldenrod and if all of the horses finish the race?

(A) 720 (B) 360 (C) 120 (D) 24 (E) 21

All 6 horses can finish the race in 6! way (assuming no tie).

If no tie is possible between No Hope and Goldenrod, then in half of these cases No Hope will be before Goldenrod and in half of these cases after (not necessarily right before or right after). How else? So, there are 6!/2=360 different arrangements of finishes where No Hope always finishes before Goldenrod.

Here's my thinking: order (from 2nd-8th place) of 6 horses does't matter. So we have 6!=720 choices 720 choices include 2 cases: No Hope comes first and Goldenrod comes first. Final answer is 720/2=360. Please correct me if there is anything wrong. Thanks

Re: Goldenrod and No Hope are in a horse race with 6 contestants [#permalink]

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07 Dec 2015, 18:29

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
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Re: Goldenrod and No Hope are in a horse race with 6 contestants [#permalink]

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27 Mar 2016, 12:20

Hi Veterans, I have a query, Since Glen and No Hope are always fixed wr to each other, cant we just combine them into an entity and thus have 5 entities left with us which can be arranged in 5! ways ? However, this gives wrong answer. Appreciate a quick reply.

Hi Veterans, I have a query, Since Glen and No Hope are always fixed wr to each other, cant we just combine them into an entity and thus have 5 entities left with us which can be arranged in 5! ways ? However, this gives wrong answer. Appreciate a quick reply.

Thanks a ton!

No Hope always finishes before Goldenrod does NOT mean that No Hope always finishes right before Goldenrod, there might be some other contestants between them.
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Re: Goldenrod and No Hope are in a horse race with 6 contestants [#permalink]

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27 Mar 2016, 12:46

Thanks Bunuel, clear as a crystal!

Bunuel wrote:

mikeonbike wrote:

Hi Veterans, I have a query, Since Glen and No Hope are always fixed wr to each other, cant we just combine them into an entity and thus have 5 entities left with us which can be arranged in 5! ways ? However, this gives wrong answer. Appreciate a quick reply.

Thanks a ton!

No Hope always finishes before Goldenrod does NOT mean that No Hope always finishes right before Goldenrod, there might be some other contestants between them.

Re: Goldenrod and No Hope are in a horse race with 6 contestants [#permalink]

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10 Jun 2016, 01:49

Hi Veterans,

I have a very embarrassing doubt,

The question stem reads "Goldenrod and No Hope are in a horse race with 6 contestants" doesn't that means we have 8 horses in the race ? (By calculating 6! and dividing it by 2 will only give stats about the 6 horses !!)

Re: Goldenrod and No Hope are in a horse race with 6 contestants [#permalink]

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24 Aug 2016, 13:05

MohitRulz wrote:

Goldenrod and No Hope are in a horse race with 6 contestants. How many different arrangements of finishes are there if No Hope always finishes before Goldenrod and if all of the horses finish the race?

(A) 720 (B) 360 (C) 120 (D) 24 (E) 21

There are 6 horses and hence 6! outcomes are possible and in that 1/2 the time No Hope will be ahead of Goldenrod and 1/2 the times lag. So it's \(\frac{6!}{2}\) = 360 (B)

I wonder if the stem is ambiguous because it only state different arrangements of finishes, i think cases in which all horses finish at the same time, or 2 horses finish at the same time, or 3 horses finish at the same time are possible.

In solutions above, they readily assume that all 6 horses finish at 6 different times.

How do you think?
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