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Goldenrod and No Hope are in a horse race with 6 contestants

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

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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?

(A) 720
(B) 360
(C) 120
(D) 24
(E) 21
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New post 11 Aug 2009, 03:16
just think of it like this

THere are 6 possible outcomes for person A to finish the race
and 5 for person B and so on

so u should multpily them all 6X5X4X...

and u get 720

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Re: GoldenRod and No Hope. [#permalink]

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combinations are symmetrical, so in half combinations goldenrod will be ahead and in exactly half it will be behind.

ans: 6!/2 = 360
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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?

(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.

Answer: B.

Similar question: six-mobsters-have-arrived-at-the-theater-for-the-premiere-of-the-126151.html
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Re: Goldenrod and No Hope are in a horse race with 6 contestants [#permalink]

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I have solved this in below fashion

Attachment:
gamtQ2.png
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)
...

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

please explain flaw in my explanation.

Thanks in Advance

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

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

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New post 23 May 2013, 08:54
I will also apply the brunel specified method.

I just wanted to know where I went wrong and correct my understanding.

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vishnuvardhan777 wrote:
I have solved this in below fashion

Attachment:
gamtQ2.png


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;

120+96+72+48+24=360.

Answer: B.

Hope it's clear.
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Re: Goldenrod and No Hope are in a horse race with 6 contestants [#permalink]

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vishnuvardhan777 wrote:
I have solved this in below fashion

Attachment:
gamtQ2.png


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


Questions about the same concept to practice:
susan-john-daisy-tim-matt-and-kim-need-to-be-seated-in-130743.html
meg-and-bob-are-among-the-5-participants-in-a-cycling-race-58095.html
in-how-many-different-ways-can-the-letters-a-a-b-91460.html
mary-and-joe-are-to-throw-three-dice-each-the-score-is-the-126407.html

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

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New post 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?

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

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New post 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.

Answer: B.

Similar question: six-mobsters-have-arrived-at-the-theater-for-the-premiere-of-the-126151.html


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

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

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New post 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.

Thanks a ton! :-D

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

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New post 27 Mar 2016, 12:41
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! :-D


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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New post 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! :-D


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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New post 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 !!)

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

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New post 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)

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

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New post 21 Aug 2017, 18:35
Hi Bunuel!

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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Re: Goldenrod and No Hope are in a horse race with 6 contestants   [#permalink] 21 Aug 2017, 18:35
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