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

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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?

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.

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:

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:

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)

We are told in the stem that only 3 medals will be awarded, so we should take this as a fact.
_________________

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)?

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;

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

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20 Dec 2014, 09:37

If this question comes, you are already at QA 60.

4C3 = 4 ways of selecting 3 people.

G S B 3 0 0 -> only 1 way as 3 identical gold medals are given to all the three people 2 1 0 -> 3 ways as GGS, GSG, SGG 1 2 0 -> 3 ways 1 1 1 -> 3! = 6 ways

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

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20 Dec 2014, 21:18

Bunuel wrote:

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:

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)

We are told in the stem that only 3 medals will be awarded, so we should take this as a fact.

Question: 1. Why do we take 'N' into consideration? why cant the patterns be : GGG, GGS, GSS, GSB? 2. considering patterns mentioned in your above explanation, why isnt GSBB a pattern too ?

Question: 1. Why do we take 'N' into consideration? why cant the patterns be : GGG, GGS, GSS, GSB? 2. considering patterns mentioned in your above explanation, why isnt GSBB a pattern too ?

1. We have 4 people in the race. 2. We are told that exactly three medals are awarded not 4.
_________________

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

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21 Dec 2014, 05:39

Bunuel wrote:

vinbitstarter wrote:

Question: 1. Why do we take 'N' into consideration? why cant the patterns be : GGG, GGS, GSS, GSB? 2. considering patterns mentioned in your above explanation, why isnt GSBB a pattern too ?

1. We have 4 people in the race. 2. We are told that exactly three medals are awarded not 4.

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

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14 Oct 2016, 22:58

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I am posting both wordy but useful explanation ( just for the understanding ) and shorter version.

Longer version:

First, let's consider the different medal combinations that can be awarded to the 3 winners: (1) If there are NO TIES then the three medals awarded are: GOLD, SILVER, BRONZE. (2) What if there is a 2-WAY tie? --If there is a 2-WAY tie for FIRST, then the medals awarded are: GOLD, GOLD, SILVER. --If there is a 2-WAY tie for SECOND, then the medals awarded are: GOLD, SILVER, SILVER. --There cannot be a 2-WAY tie for THIRD (because exactly three medals are awarded intotal). (3) What if there is a 3-WAY tie? --If there is a 3-WAY tie for FIRST, then the medals awarded are: GOLD, GOLD, GOLD. --There are no other possible 3-WAY ties. Thus, there are 4 possible medal combinations: (1) G, S, B (2) G, G, S (3) G, S, S (4) G, G, G. Now let's determine how many different ways each combination can be distributed. We'll do this by considering four runners: Albert, Bob, Cami, and Dora.

COMBINATION 1: Gold, Silver, Bronze Gold Medal Silver Medal Bronze Medal Any of the 4 runners can receive the gold medal. There are only 3 runners who can receive the silver medal. Why? One of the runners has already been awarded the Gold Medal. There are only 2 runners who can receive the bronze medal. Why? Two of the runners have already been awarded the Gold and Silver medals. 4 possibilities 3 possibilities 2 possibilities Therefore, there are different victory circles that will contain 1 GOLD, 1 SILVER, and 1 BRONZE medalist.

COMBINATION 2: Gold, Gold, Silver. Using the same reasoning as for Combination 1, we see that there are 24 different victory circles that will contain 2 GOLD medalists and 1 SILVER medalist. However, it is important to realize that these 24 victory circles must be reduced due to "overcounting." To illustrate this, consider one of the 24 possible Gold-GoldSilver victory circles: Albert is awarded a GOLD. Bob is awarded a GOLD. Cami is awarded a SILVER. Notice that this is the exact same victory circle as the following: Bob is awarded a GOLD. Albert is awarded a GOLD. Cami is awarded a SILVER. Each victory circle has been "overcounted" because we have been counting each different ordering of the two gold medals as a unique victory circle, when, in reality, the two different orderings consist of the exact same victory circle. Thus, the 24 victory circles must be cut in half; there are actually only 12unique victory circles that will contain 2 GOLD medalists and 1 SILVER medalist. (Note that we did not have to worry about "overcounting" in Combination 1, because each of those 24 possibilities was unique.)

COMBINATION 3: Gold, Silver, Silver. Using the same reasoning as for Combination 2, we see that there are 24 possible victory circles, but only 12 unique victory circles that contain 1 GOLD medalist and 2 SILVER medalists.

COMBINATION 4: Gold, Gold, Gold.

Here, once again, there are 24 possible victory circles. However, because all three winners are gold medalists, there has been a lot of "overcounting!" How much overcounting? Let's consider one of the 24 possible GoldGold-Gold victory circles: Albert is awarded a GOLD. Bob is awarded a GOLD. Cami is awarded a GOLD. Notice that this victory circleis exactly the same as the following victory circles: Albert-GOLD, CamiGOLD, Bob-GOLD. Bob-GOLD, Albert-GOLD, Cami-GOLD. Bob-GOLD, Cami-GOLD, Albert-GOLD. Cami-GOLD, Albert-GOLD, Bob-GOLD. Cami-GOLD, Bob-GOLD, Albert-GOLD. Each unique victory circlehas actually been counted 6 times! Thus we must divide 24 by 6 to find the number of unique victory circles. There are actually only unique victory circlesthat contain 3 GOLD medalists. FINALLY, then, we have the following:

(Combination 1) 24 unique GOLD-SILVER-BRONZE victory circles. (Combination 2) 12 unique GOLDGOLD-SILVER victory circles. (Combination 3) 12 unique GOLD-SILVER-SILVER victory circles. (Combination 4) 4 unique GOLD-GOLD-GOLD victory circles. Thus, there are uniquevictory circles. The correct answer is B.

Shorter version:

Case 1: G+S+B --> 4P3 = 4C3 3! = 24 Case 2: G+G+S --> 4C2 * 2C1 = 12 Case 3: G+S+S --> 4C2 * 2C1 = 12 Case 4: G+G+G --> 4C3 = 4

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

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03 Jan 2017, 20:21

Bunuel wrote:

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.

I understand complete solution except this part "4. Gold/Gold/Gold/No medal - 4!/3!=4." Why we are considering this case?
_________________

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.

I understand complete solution except this part "4. Gold/Gold/Gold/No medal - 4!/3!=4." Why we are considering this case?

This is a case when 3 runners are tied for gold medal.
_________________

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

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30 Mar 2017, 21:36

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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

OFFICIAL SOLUTION

First, let's consider the different medal combinations that can be awarded to the 3 winners:

(1) If there are NO TIES then the three medals awarded are: GOLD, SILVER, BRONZE.

(2) What if there is a 2-WAY tie? --If there is a 2-WAY tie for FIRST, then the medals awarded are: GOLD, GOLD, SILVER. --If there is a 2-WAY tie for SECOND, then the medals awarded are: GOLD, SILVER, SILVER. --There cannot be a 2-WAY tie for THIRD (because exactly three medals are awarded in total).

(3) What if there is a 3-WAY tie? --If there is a 3-WAY tie for FIRST, then the medals awarded are: GOLD, GOLD, GOLD. --There are no other possible 3-WAY ties.

Thus, there are 4 possible medal combinations:

(1) G, S, B (2) G, G, S (3) G, S, S (4) G, G, G

Now let's determine how many different ways each combination can be distributed. We'll do this by considering four runners: Albert, Bob, Cami, and Dora.

COMBINATION 1: Gold, Silver, Bronze

Gold Medal Any of the 4 runners can receive the gold medal. (4 possibilities) Silver Medal There are only 3 runners who can receive the silver medal. Why? One of the runners has already been awarded the Gold Medal. (3 possibilities) Bronze Medal There are only 2 runners who can receive the bronze medal. Why? Two of the runners have already been awarded the Gold and Silver medals. (2 possibilities)

Therefore, there are \(4*3*2=24\) different victory circles that will contain 1 GOLD, 1 SILVER, and 1 BRONZE medalist.

COMBINATION 2: Gold, Gold, Silver.

Using the same reasoning as for Combination 1, we see that there are 24 different victory circles that will contain 2 GOLD medalists and 1 SILVER medalist. However, it is important to realize that these 24 victory circles must be reduced due to "overcounting."

To illustrate this, consider one of the 24 possible Gold-Gold-Silver victory circles:

Albert is awarded a GOLD. Bob is awarded a GOLD. Cami is awarded a SILVER.

Notice that this is the exact same victory circle as the following:

Bob is awarded a GOLD. Albert is awarded a GOLD. Cami is awarded a SILVER.

Each victory circle has been "overcounted" because we have been counting each different ordering of the two gold medals as a unique victory circle, when, in reality, the two different orderings consist of the exact same victory circle. Thus, the 24 victory circles must be cut in half; there are actually only 12 unique victory circles that will contain 2 GOLD medalists and 1 SILVER medalist. (Note that we did not have to worry about "overcounting" in Combination 1, because each of those 24 possibilities was unique.)

COMBINATION 3: Gold, Silver, Silver.

Using the same reasoning as for Combination 2, we see that there are 24 possible victory circles, but only 12 unique victory circles that contain 1 GOLD medalist and 2 SILVER medalists.

COMBINATION 4: Gold, Gold, Gold.

Here, once again, there are 24 possible victory circles. However, because all three winners are gold medalists, there has been a lot of "overcounting!" How much overcounting?

Let's consider one of the 24 possible Gold-Gold-Gold victory circles:

Albert is awarded a GOLD. Bob is awarded a GOLD. Cami is awarded a GOLD.

Notice that this victory circle is exactly the same as the following victory circles:

Each unique victory circle has actually been counted 6 times! Thus we must divide 24 by 6 to find the number of unique victory circles. There are actually only \(\frac{24}{6}=4\) unique victory circles that contain 3 GOLD medalists.

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?

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

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05 May 2017, 08:48

One of the key things mentioned here is " they stand in a victory circle" so shouldn't we multiple 52 with 2!, for making an appropriate circle arrangement in each case.