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Math Help - 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
    Super Member 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 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 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 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: . \text{7 choices}

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


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

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