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Math Help - number of ways

  1. #1
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    number of ways

    2 3 4 5 6 7 8 9 }


    in how many ways you can obtain 4 pairs from the following with no pair having a GCD 2
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  2. #2
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    Hello, dhruviboy!

    { 2 3 4 5 6 7 8 9 }

    In how many ways you can obtain 4 pairs with no pair having a GCD 2?
    A pair has GCD 2 if both numbers are even.
    . . Hence, we must form pairs in which one is odd and one is even.


    We have the four odd numbers: . 3\;\;5\;\;7\;\;9

    In how many ways can the even numbers be assigned?

    The "2" can be paired with any of the 4 odd numbers.
    Then the "4" can be paired with any of the remaining 3 odd numbers.
    Then the "6" can be paired with either of the remaining 2 odd numbers.
    And the "8" is paired with the 1 remaining odd number.

    Therefore, there are: . 4! \:=\:{\color{blue}24} possible pairings.

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

    A pair has GCD 2 if both numbers are even.
    . . Hence, we must form pairs in which one is odd and one is even.


    We have the four odd numbers: . 3\;\;5\;\;7\;\;9

    In how many ways can the even numbers be assigned?

    The "2" can be paired with any of the 4 odd numbers.
    Then the "4" can be paired with any of the remaining 3 odd numbers.
    Then the "6" can be paired with either of the remaining 2 odd numbers.
    And the "8" is paired with the 1 remaining odd number.

    Therefore, there are: . 4! \:=\:{\color{blue}24} possible pairings.
    sir you have quoted"A pair has GCD 2 if both numbers are even". then please tell what is the gcd of 4 and 8.
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