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Thread: Permutations & Organized Counting

  1. #1
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    Permutations & Organized Counting

    Hello, I would like some help in solving these problems. Please explain how you arrived at your answer if you don't mind.

    1. Ten finalists are competing in a race at the Canada games:
    a) In how many different orders can the competitors finish the race?
    b) How many ways could the gold, silver, and bronze medal be awarded?
    c) One of the finalists is a friend from your town. How many of the possible finishes would include them winning a medal?
    d) How many possible finishes would leave your friend out of the medal standings?

    2. The final score of a soccer game is 6 to 3. How many different scores were possible at half-time?

    Thanks for the help!
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  2. #2
    MHF Contributor Matt Westwood's Avatar
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    a) There are 10 possible winners.

    For each of those 10 winners, there are 9 possible runners who could come 2nd.

    For each of those 10 winners and 9 second-placers, there are 8 possible runners who can come 3rd.

    And so on.

    Thus there are $\displaystyle 10 \times 9 \times 8 \times \ldots \times 2 \times 1$ different orders that the race can finish in (if you discard the possibility of a dead heat between two finishers).

    This is written $\displaystyle 10!$ "10 factorial" and it's about 3.2 million.

    b) When you're interested only in gold, silver and bronze you just need the first three, that is $\displaystyle 10 \times 9 \times 8$.

    That'll do for a start, hope it gives you a clue about how to continue.
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  3. #3
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    Hello, NineZeroFive!

    2. The final score of a soccer game is 6 to 3.
    How many different scores were possible at half-time?

    Team A (winning team) could have had any score from 0 to 6: .$\displaystyle \text{7 choices}$

    Team B could have had any score from 0 to 3: .$\displaystyle 4\text{ choices}$


    Therefore, there were: .$\displaystyle 7 \times 4 \:=\:28$ possible half-time scores.

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