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# There are 19 figure skaters in the Olympic women's competition, including 3 Americans. The gold medal goes to first place, silver to second,

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There are 19 figure skaters in the Olympic women's competition, including 3 Americans. The gold medal goes to first place, silver to second, and bronze to third.

Suppose that the Olympics powers-that-be decide that exactly one American must win a medal (no more and no less). Now how many ways are there to award the medals?

Guest Feb 20, 2015

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Hi Geno, I was looking at what you did.

If you choose the non US prize winners like you did that is 16*15 then you are saying that there are 240 permutaions of those 2 places.

then 1 us person must be chosen.  this can be done in 3 ways

now that US person can be 1st 2nd or 3rd     So 3 ways

240*3*3 = 2160

You should not multiply this by 2 as all the possible permutations of the non US prize winners in relation to each other have already been accounted for.

Melody  Feb 21, 2015
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There are three different choices for the American.

There are 16 different choices for the first non-American and 15 different choices for the other non-American.

So, there are 3 x 16 x 15 different groups.

But, order is important -- who gets the gold, who gets the silver, and who gets the bronze -- there are 6 different ways to arrange the 3 winners (3! = 6).

Thus, there are 6 x 3 x 16 x 15 different results.

geno3141  Feb 20, 2015
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I know my answer is incorrect and geno's is correct.....but can someone spot my error??

I chose to see the problem this way.....

In each set of three people, we want to choose any 1 of 3 Americans and then choose any 2 of 16 of the other contestants

So the total possible sets - disregarding order, is  C(3,1)*C(16,2) = 3 * 120  = 360

And each of these sets can be ordered in 3!  = 6 ways

So......6 x 360 = 2160  which is 1/2 of geno's result.......????....where did i go wrong???

CPhill  Feb 20, 2015
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CPhill -- I think the problem arises by choosing two groups, the groups of Americans and the groups of non-Americans.

By doing the problem this way, there are two arrangements, either choosing the Americans first and then the non-Americans, or vice-versa. Then, there are the six arrangements of the individuals. This would double your answer.

Or, maybe not ... ?

geno3141  Feb 21, 2015
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Why do you think that you are wrong Chris ? I think that yours is the correct solution.

Geno's solution is wrong because it orders the non-Americans twice. The 16×15 is the number of permutations of 2 from 16 and so is already taking order into account. If simply two non-Americans were required, then C (16, 2)=16×15/2 should be used.

Here's an alternative way of arriving at the result

The number of ways in which America wins a single medal, gold = 3×16×15 = 720.

The number of ways in which America wins a single medal, silver = 16×3×15 = 720.

The number of ways in which America wins a single medal, bronze = 16×15×3 = 720.

The total is 720 + 720 + 720 = 2160.

Guest Feb 21, 2015
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OK...thanks, Melody....you know....I'm NEVER too sure about my answers to any probability question....!!!!

CPhill  Feb 21, 2015
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Why are you talking like I wrote that last annonymous comment Chris?

I did not write that.

However,

I just looked at the question and I agree with you and anon.  That is how I would have done it.

I am not saying it is necessarily correct but it does make sense to me.

Melody  Feb 21, 2015
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Sorry....I misread that one.......maybe I'm still dizzy....LOL!!!

CPhill  Feb 21, 2015
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I do not think that anyone on this forum is really strong when it comes to probability.

I do however think that your knowledge has been improving very rapidly in this subject area.  Now all you really need is a bit more confidence.

Melody  Feb 21, 2015
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....Or a bit more wine......

CPhill  Feb 21, 2015
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Yes that might help too - might help me too LOL  :))

Melody  Feb 21, 2015
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Hi Geno, I was looking at what you did.

If you choose the non US prize winners like you did that is 16*15 then you are saying that there are 240 permutaions of those 2 places.

then 1 us person must be chosen.  this can be done in 3 ways

now that US person can be 1st 2nd or 3rd     So 3 ways

240*3*3 = 2160

You should not multiply this by 2 as all the possible permutations of the non US prize winners in relation to each other have already been accounted for.

Melody  Feb 21, 2015

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