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# Counting

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help how do you do this thing

In the olympic women's skating competition, the gold medal goes to first place, silver to second, and bronze to third.  If there are skaters, including Americans, in how many ways can the medals be awarded to three of the skaters if exactly one of the americans wins a medal?

Mar 6, 2023

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We can solve this problem using the multiplication principle of counting.

First, we need to choose one American skater to win a medal. There are 'n' ways to do this, where 'n' is the number of American skaters.

Next, we need to choose two skaters from the remaining non-American skaters to win the other two medals. There are (n-1) ways to do this, since we have already chosen one American skater to win a medal. We can choose the first non-American skater to win a medal in (n-1) ways, and then choose the second non-American skater to win a medal in (n-2) ways (since there is one less non-American skater remaining).

Therefore, the total number of ways to award the medals, given that exactly one American skater wins a medal, is:

n * (n-1) * (n-2)

Note that this expression assumes that there are at least three skaters competing, and that there is at least one American skater. If either of these assumptions is not true, the expression will need to be modified accordingly.

Mar 6, 2023