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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] New post 03 Dec 2009, 06:52
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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
[Reveal] Spoiler: OA
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Re: race [#permalink] New post 03 Dec 2009, 09:42
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kirankp 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


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

Answer: B.
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Re: race [#permalink] New post 06 Apr 2010, 02:44
Quote:
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

Answer: B.



Sorry, but i dont see how this answers the "how many different victory circles are possible" part of the question?


If we are concerned only with medals in the victory circle:

Possible medal combinations are: GSB, GSS, GGS, GGG.
Possible victory circles [N.B. removing rotationally symmetric permutations]:

GSB: = 2
GSS: = 1
GGS: = 1
GGG: = 1

So, total possible victory circles are 5.


If we are concerned with combinations of people and medals in the victory circle:


Possible permutation of winners (top 3) = 4P3 = 24
Possible victory circles:

GSB: = 24*2 = 48
GSS: = 48 [N.B. S1 <> S2]
GGS: = 48
GGG: = 48

So, total possible victory circles are 192... :?:

Please can i get some advice on where i went wrong with both of these approaches? :?
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Re: race [#permalink] New post 06 Apr 2010, 22:31
what if 2 people come first(i.e. Gold) and
another 2 come second position(silver) :-D
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Re: race [#permalink] New post 07 Apr 2010, 00:45
Expert's post
Ramsay wrote:
Quote:
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

Answer: B.



Sorry, but i dont see how this answers the "how many different victory circles are possible" part of the question?


If we are concerned only with medals in the victory circle:

Possible medal combinations are: GSB, GSS, GGS, GGG.
Possible victory circles [N.B. removing rotationally symmetric permutations]:

GSB: = 2
GSS: = 1
GGS: = 1
GGG: = 1

So, total possible victory circles are 5.


If we are concerned with combinations of people and medals in the victory circle:


Possible permutation of winners (top 3) = 4P3 = 24
Possible victory circles:

GSB: = 24*2 = 48
GSS: = 48 [N.B. S1 <> S2]
GGS: = 48
GGG: = 48

So, total possible victory circles are 192... :?:

Please can i get some advice on where i went wrong with both of these approaches? :?


I don't quite understand your solution. Here is the logic behind mine:

We have four possible patterns (GSBN, GSSN, GGSN, GGGN) and four persons (a, b, c, d). Each victory circle is made by assigning pattern letters to these persons as follows:
Firs pattern: GSBN
a--b--c--d
G--S--B--N
G--B--S--N
G--B--N--S
...
So how many victory circles are possible for the first pattern? It would be the # of permutations of the letters GSBN, as these are four distinct letters, the # is 4!=24.

The same for other patterns:
Second pattern: GSSN. # of permutations is 4!/2!=12 (as there are four letters out of which 2 are the same).

Third pattern: GGSN. # of permutations is 4!/2!=12.

Fourth pattern: GGGN. # of permutations is 4!/3!=4.
a--b--c--d
G--G--G--N
G--G--N--G
G--N--G--G
N--G--G--G
Here you can see that these four victory circles are all different and the same will be for other patterns.

24+12+12+4=52

Hope it's clear.

RaviChandra wrote:
what if 2 people come first(i.e. Gold) and
another 2 come second position(silver) :-D


We are told in the stem that only 3 medals will be awarded, so we should take this as a fact.
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Re: race [#permalink] New post 07 Apr 2010, 01:30
Bunuel,

Thanks for the clarification - I'm still not comfortable with the answer and i have three more questions:

1) Is a "victory circle" an actual circle and therefore are there rotational symmetry considerations that need to be taken into account when calculating arrangements. [I've never heard of this term before - so apologies if this is a silly question?]

2) The question asks: "three medal winners stand together with their medals to form a victory circle" so are we calculating permutations with to many people in the 'victory circle'? [This is the only flaw i'm pretty sure about which takes the number of permutation down to 13 - which is not an answer]

3)The questions asks: "the three medal winners stand together with their medals" so should we be calculating permutations of people and medals and not discounting duplicate medals? [Potentially tenuous but needed to get to an answer - see below]


Assuming that (1) a 'victory circle' is not a circle but a straight line, (2) that we are only concerned with medal holders, and (3) person A with a silver medal can be considered different to person B with a silver medal (in a scenario where they both have one) then:

Possible medal combinations: GSB, GSS, GGS, GGG

GSB victory circle permutations = 4P3 = 24
GSS victory circle permutations = 4P3 = 24
GGS victory circle permutations = 4P3 = 24
GGG victory circle permutations = 4P3 = 24

Total possible permutations of victory circle = 24*4 = 96 (Answer C)? :?

EDIT: for clarity
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Re: race [#permalink] New post 07 Apr 2010, 02:41
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Ramsay wrote:
Bunuel,

Thanks for the clarification - I'm still not comfortable with the answer and i have three more questions:

1) Is a "victory circle" an actual circle and therefore are there rotational symmetry considerations that need to be taken into account when calculating arrangements. [I've never heard of this term before - so apologies if this is a silly question?]

2) The question asks: "three medal winners stand together with their medals to form a victory circle" so are we calculating permutations with to many people in the 'victory circle'? [This is the only flaw i'm pretty sure about which takes the number of permutation down to 13 - which is not an answer]

3)The questions asks: "the three medal winners stand together with their medals" so should we be calculating permutations of people and medals and not discounting duplicate medals? [Potentially tenuous but needed to get to an answer - see below]


Assuming that (1) a 'victory circle' is not a circle but a straight line, (2) that we are only concerned with medal holders, and (3) person A with a silver medal can be considered different to person B with a silver medal (in a scenario where they both have one) then:

Possible medal combinations: GSB, GSS, GGS, GGG

GSB victory circle permutations = 4P3 = 24
GSS victory circle permutations = 4P3 = 24
GGS victory circle permutations = 4P3 = 24
GGG victory circle permutations = 4P3 = 24

Total possible permutations of victory circle = 24*4 = 96 (Answer C)? :?

EDIT: for clarity


OK, first of all victory circle is not the actual circle. Question asks how many different scenarios are possible for: medal-person; medal-person; medal-person.

Let's consider the easiest one - scenario GGG. You say there are 24 circles (scenarios) possible, but there are only 4:

1. G/a, G/b, G/c (here G/b, G/a, G/c is the same scenario a, b and c won the gold);
2. G/a, G/b, G/d;
3. G/a, G/c, G/d;
4. G/b, G/c, G/d.

Scenario GGS:
1. G/a, G/b, S/c (a and b won gold and c won silver. Here G/b, G/a, S/c is the same scenario: a and b won gold and c won silver;
2. G/a, G/b, S/d;
3. G/a, G/c, S/b;
4. G/a, G/c, S/d;
5. G/a, G/d, S/b;
6. G/a, G/d, S/c;

7. G/b, G/c, S/a;
8. G/b, G/c, S/d;
9. G/b, G/d, S/a;
10. G/b, G/d, S/c;

11. G/c, G/d, S/a;
12. G/c, G/d, S/b.

You can see that all 12 scenarios are different in terms of medal/person, medal/person, medal/person. The same will be for GSS.

For GSB you are right there will be 4P3=24 different scenarios, or as I wrote 4!=24.

So again 24(GSB)+12(GGS)+12(GSS)+4(GGG)=52 (which is OA according to kirankp).

Hope it's clear.
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Re: race [#permalink] New post 07 Apr 2010, 05:59
Thanks! I didn't understand the approach end to end but that's a fantastic explanation. +1
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Re: race [#permalink] New post 09 Apr 2010, 22:58
super explanation ...
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Re: In a 4 person race, medals are awarded to the fastest 3 runn [#permalink] New post 16 Aug 2014, 09:00
I get that there are 52 arrangements of medals and people, but

if they stand in a circle of 3 people, there are 2 circles that can be formed per arrangement above

so the total number of possible victory circles is 104

What do you think?
Re: In a 4 person race, medals are awarded to the fastest 3 runn   [#permalink] 16 Aug 2014, 09:00
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