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# In a 4 person race, medals are awarded to the fastest 3 runn

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In a 4 person race, medals are awarded to the fastest 3 runn [#permalink]

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18 Sep 2007, 23:32
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95% (hard)

Question Stats:

38% (02:23) correct 62% (01:23) wrong based on 173 sessions

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In a 4 person race, medals are awarded to the fastest 3 runners. The first-place runner receives a gold medal, the second-place runner receives a silver medal, and the third-place runner receives a bronze medal. In the event of a tie, the tied runners receive the same color medal. (For example, if there is a two-way tie for first-place, the top two runners receive gold medals, the next-fastest runner receives a silver medal, and no bronze medal is awarded). Assuming that exactly three medals are awarded, and that the three medal winners stand together with their medals to form a victory circle, how many different victory circles are possible?

A. 24
B. 52
C. 96
D. 144
E. 648

OPEN DISCUSSION OF THIS QUESTION IS HERE: in-a-4-person-race-medals-are-awarded-to-the-fastest-3-runn-87569.html
[Reveal] Spoiler: OA

Last edited by Bunuel on 01 Oct 2013, 01:09, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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19 Sep 2007, 00:20
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12345678 wrote:
In a 4 person race, medals are awarded to the fastest 3 runners. The first-place runner receives a gold medal, the second-place runner receives a silver medal, and the third-place runner receives a bronze medal. In the event of a tie, the tied runners receive the same color medal. (For example, if there is a two-way tie for first-place, the top two runners receive gold medals, the next-fastest runner receives a silver medal, and no bronze medal is awarded). Assuming that exactly three medals are awarded, and that the three medal winners stand together with their medals to form a victory circle, how many different victory circles are possible?

a)24
b)52
c)96
d)144
e)648

wtf is a victory circle lol??? does this mean that all the medals are the same??? screw them and their stupid medals!

I don't like this one
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19 Sep 2007, 02:49
12345678 wrote:
In a 4 person race, medals are awarded to the fastest 3 runners. The first-place runner receives a gold medal, the second-place runner receives a silver medal, and the third-place runner receives a bronze medal. In the event of a tie, the tied runners receive the same color medal. (For example, if there is a two-way tie for first-place, the top two runners receive gold medals, the next-fastest runner receives a silver medal, and no bronze medal is awarded). Assuming that exactly three medals are awarded, and that the three medal winners stand together with their medals to form a victory circle, how many different victory circles are possible?

a)24
b)52
c)96
d)144
e)648

Gold can be awarded in 4 ways, then silver can be awarded in 3 ways and Bronze in 2 ways.

therefore the total number of ways are : 4*3*2=24ways of awarding the medals and the same number of ways of forming the circle.

This is when there is no tie. And if there is tie, for example all three receive the GOLD or the Silver or the Bronze, then there are 4 more cases. which implies 24*4=96.

I guess I am still missing something. Good question!

What is the OA?
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19 Sep 2007, 11:55
I also ask - what is a victory circle? I've never heard that term before. Is it even relevant to the problem?
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19 Sep 2007, 19:23
LM wrote:
12345678 wrote:
In a 4 person race, medals are awarded to the fastest 3 runners. The first-place runner receives a gold medal, the second-place runner receives a silver medal, and the third-place runner receives a bronze medal. In the event of a tie, the tied runners receive the same color medal. (For example, if there is a two-way tie for first-place, the top two runners receive gold medals, the next-fastest runner receives a silver medal, and no bronze medal is awarded). Assuming that exactly three medals are awarded, and that the three medal winners stand together with their medals to form a victory circle, how many different victory circles are possible?

a)24
b)52
c)96
d)144
e)648

Gold can be awarded in 4 ways, then silver can be awarded in 3 ways and Bronze in 2 ways.

therefore the total number of ways are : 4*3*2=24ways of awarding the medals and the same number of ways of forming the circle.

This is when there is no tie. And if there is tie, for example all three receive the GOLD or the Silver or the Bronze, then there are 4 more cases. which implies 24*4=96.

I guess I am still missing something. Good question!

What is the OA?

I took it as far as you. but then what about ties where 2 people tie for one medal and the 3rd person wins the other?

choices for gold = 4, choices for other gold = 3, choices for silver = 2. 4*3*2 = 24.

choices for gold = 4, choices for silver = 3, choices for other silver = 2. 4*3*2 = 24

so 96 + 24 + 24 = 144

I get D for this one.

do we have an OA?
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20 Sep 2007, 04:28
12345678 wrote:
In a 4 person race, medals are awarded to the fastest 3 runners. The first-place runner receives a gold medal, the second-place runner receives a silver medal, and the third-place runner receives a bronze medal. In the event of a tie, the tied runners receive the same color medal. (For example, if there is a two-way tie for first-place, the top two runners receive gold medals, the next-fastest runner receives a silver medal, and no bronze medal is awarded). Assuming that exactly three medals are awarded, and that the three medal winners stand together with their medals to form a victory circle, how many different victory circles are possible?

a)24
b)52
c)96
d)144
e)648

There are 4 possible combinations:
123
111
112
122

and for each combination:
for the first digit rival 4 people
for the second digit rival 3 people
for the third digit rival 2 people

Therefore for each combination we have 4*3*2=24
Total=4combinations*24=96
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22 Sep 2009, 09:47
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This is a Manhattan GMAT question. OA is B.

It can be found on "Manhattan GMAT Archive (tough problems set).doc" and on http://www.manhattangmat.com/forums/gma ... -t886.html.

I don't like their answer; too Manhattan like = too wordy.

Scenario 1: g+s+b --> 4P3 = 4C3 3! = 24
Scenario 2: g+g+s --> 4C2 * 2C1 = 12
Scenario 3: g+s+s --> 4C2 * 2C1 = 12
Scenario 4: g+g+g --> 4C3 = 4

24+12+12+4=52

This answer assumes that the order of the runners receiving same medals, which is random, does not affect the positioning on the victory circle.
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02 May 2011, 09:28
Interesting numerical.
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Re: In a 4 person race, medals are awarded to the fastest 3 [#permalink]

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30 Sep 2013, 12:44
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Re: In a 4 person race, medals are awarded to the fastest 3 [#permalink]

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30 Sep 2013, 21:16
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12345678 wrote:
In a 4 person race, medals are awarded to the fastest 3 runners. The first-place runner receives a gold medal, the second-place runner receives a silver medal, and the third-place runner receives a bronze medal. In the event of a tie, the tied runners receive the same color medal. (For example, if there is a two-way tie for first-place, the top two runners receive gold medals, the next-fastest runner receives a silver medal, and no bronze medal is awarded). Assuming that exactly three medals are awarded, and that the three medal winners stand together with their medals to form a victory circle, how many different victory circles are possible?

a) 24
b) 52
c) 96
d) 144
e) 648

A victory circle is where winners stand together in any formation. The reason victory circle is used here is that the question does not ask you in how many ways can you have different winners. It asks you in how many ways can you have different winners or different medals. e.g. if there are 4 runners A, B, C and D, and if A, B and C are winners,
A - Gold, B - Silver, C - Bronze is different from A - Gold, B - Gold, C - Gold.
Hence they have included runners as well as their medals in the victory circle.

In how many ways can medals be awarded?

Attachment:

Ques3.jpg [ 17.98 KiB | Viewed 8558 times ]

There are 4 ways:
GGG - Select 3 of the 4 people - 4C3 = 4
GGS - Select 3 people and arrange the medals among them - 4C3 * 3!/2! = 12
GSS - Select 3 people and arrange the medals among them - 4C3 * 3!/2! = 12
GSB - Select 3 people and arrange the medals among them - 4C3 * 3! = 24

Total different victory circles = 4+12+12+24 = 52
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Re: In a 4 person race, medals are awarded to the fastest 3 runn [#permalink]

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01 Oct 2013, 01:10
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In a 4 person race, medals are awarded to the fastest 3 runners. The first-place runner receives a gold medal, the second-place runner receives a silver medal, and the third-place runner receives a bronze medal. In the event of a tie, the tied runners receive the same color medal. (For example, if there is a two-way tie for first-place, the top two runners receive gold medals, the next-fastest runner receives a silver medal, and no bronze medal is awarded). Assuming that exactly three medals are awarded, and that the three medal winners stand together with their medals to form a victory circle, how many different victory circles are possible?

A. 24
B. 52
C. 96
D. 144
E. 648

Possible scenarios are:

1. Gold/Silver/Bronze/No medal (no ties) - 4!=24;
2. Gold/Gold/Silver/No medal - 4!/2!=12;
3. Gold/Silver/Silver/No medal - 4!/2!=12;
4. Gold/Gold/Gold/No medal - 4!/3!=4.

Total: 24+12+12+4=52

OPEN DISCUSSION OF THIS QUESTION IS HERE: in-a-4-person-race-medals-are-awarded-to-the-fastest-3-runn-87569.html
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Re: In a 4 person race, medals are awarded to the fastest 3 runn   [#permalink] 01 Oct 2013, 01:10
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