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Math Help - Counting Problem

  1. #1
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    Counting Problem

    Say there is a CD player that holds 5 CDs. There are 5 trays that hold one cd each. I have 100 CD's.

    In how many ways can the CD player be loaded if AT MOST two CD's are placed in it?
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  2. #2
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    Hello, jzellt!

    There is a CD player that holds 5 CDs.
    There are 5 trays that hold one CD each. I have 100 CD's.

    In how many ways can the CD player be loaded if AT MOST two CD's are placed in it?
    I assume that the 100 CD's are distinguishable
    . . and that the 5 trays are also distinguishable.


    Place one CD into the player.
    There are 100 choices for the CD.
    There are 5 choices for the tray to place it in.
    . . There are: . 100 \times 5 \:=\:500 ways to load one CD into the player.


    Place two CD's into the player.
    There are {100\choose2} \:=\:4950 ways to choose two CD's.
    There are: . _5P_2 \:=\:20 ways to place them in the trays.
    . . There are: . 4950 \times 20 \:=\:99,000 ways to load two CD's into the player.


    Therefore, there are: . 500 + 99,000 \:=\:{\bf{\color{blue}99,500}} ways to load at most two CD's.

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