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Math Help - In a new lotto game, you must choose six numbers from the numbers 1 to 25. Six winn

  1. #1
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    In a new lotto game, you must choose six numbers from the numbers 1 to 25. Six winn

    In a new lotto game, you must choose six numbers from the numbers 1 to 25. Six winning numbers are drawn, then one supplemental number is drawn from the remaining numbers. You win Fifth Prize is the six numbers you have chosen contain exactly three of the winning numbers and the supplemental number. Find the probability of winning Fifth Prize.
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  2. #2
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    Hello, help1!

    In a new lotto game, you must choose six numbers from the numbers 1 to 25.
    Six winning numbers are drawn,
    . . then one supplemental number is drawn from the remaining numbers.
    You win Fifth Prize if your 6 numbers contain exactly 3 of the winning numbers
    . . and the supplemental number.

    Find the probability of winning Fifth Prize.

    There are: . {25\choose6}\:=\:177,100 ways to choose the 6 numbers
    . . and 19 choices for the supplemental number.

    Hence, there are: . 177,100 \times 19 \:=\:{\color{blue}3,364,900} possible outcomes.



    To win Fifth Prize:

    You must have 3 of the 6 winning numbers: . {6\choose3} \:=\:20 ways.

    You must have the supplemental number: . 1 way.

    Your other two numbers are from the 18 "losers": . {18\choose2}\:=\:153 ways.

    Hence, you have: . 20 \times 1 \times 153 \:=\:{\color{blue}3,060} ways of winning.


    Therefore: . P(\text{5th Prize}) \;=\;\frac{3,060}{3,364,900} \;=\;\frac{153}{168,245}

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  3. #3
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    Quote Originally Posted by Soroban View Post
    Hello, help1!


    There are: . {25\choose6}\:=\:177,100 ways to choose the 6 numbers
    . . and 19 choices for the supplemental number.

    Hence, there are: . 177,100 \times 19 \:=\:{\color{blue}3,364,900} possible outcomes.



    To win Fifth Prize:

    You must have 3 of the 6 winning numbers: . {6\choose3} \:=\:20 ways.

    You must have the supplemental number: . 1 way.

    Your other two numbers are from the 18 "losers": . {18\choose2}\:=\:153 ways.

    Hence, you have: . 20 \times 1 \times 153 \:=\:{\color{blue}3,060} ways of winning.


    Therefore: . P(\text{5th Prize}) \;=\;\frac{3,060}{3,364,900} \;=\;\frac{153}{168,245}

    What do those (25 6) mean? How do I show that = 177,100? What button on my calc does that?
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    I think it is

    Quote Originally Posted by help1 View Post
    What do those (25 6) mean? How do I show that = 177,100? What button on my calc does that?
    A permutation so it would be equivalent to \frac{25!}{6!(25-6)!} where ! denotes factorial...and you might have a button...it would look like this _nP_r
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