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805+ (Hard)|   Combinations|      
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ZulfiquarA
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In a 4 person race, medals are awarded to the fastest 3 runners. The 27 May 2021, 03:25
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We have 4 racers and 3 medals
Conditions people can tie for a medal but only 3 winners would be there

Scenario 1: No ties
4*3*2=24
Scenario 2: 2 people tie for gold
4*3=12
Scenario 3: 2 people tie for silver
3*4=12
Scenario 4: 3 people tie for gold
4
Total of scenarios=24+12+12+4=52
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puneetfitness
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In a 4 person race, medals are awarded to the fastest 3 runners. The 21 Nov 2022, 00:01
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Bunuel
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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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Hi Bunnel in case of GSB fourth person will not bt part of victory circle hence it should be 3! As victory circle should include only winners Ways for GSB not 4! Please clarify
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In a 4 person race, medals are awarded to the fastest 3 runners. The 21 Nov 2022, 00:15
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puneetfitness
Bunuel
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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.

Posted from my mobile device


Hi Bunnel in case of GSB fourth person will not bt part of victory circle hence it should be 3! As victory circle should include only winners Ways for GSB not 4! Please clarify
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We have 4 person race. So, we are giving 3 medals, and no medal (N) for the fourth person in the race, so it's basically GSBN: 4!.
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In a 4 person race, medals are awarded to the fastest 3 runners. The 19 Aug 2024, 01:24
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Its easy to miss GGG here.

I got it 4C3*(GSB+GGS+GSS+GG) = 4*(6+3+3+1)=52

Anyway the circle threw me into a loop.

It got me thinking what happens when circular permutations have repetition.

Is it (n-1)!/(r1!r2!..rn!) where r is repeated?

Doesnt feel that simple to me.
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In a 4 person race, medals are awarded to the fastest 3 runners. The 19 Aug 2024, 07:16
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When you search Google and Wikipedia as I did and see no reference to Victory circle within the limits of one's patience and then use AI to eventually find the definition, you know you're reading an inartfully crafted question

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In a 4 person race, medals are awarded to the fastest 3 runners. The 08 Sep 2025, 06:26
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