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Math Help - Why permutation ?

  1. #1
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    Why permutation ?

    we have 4 distinct ice cream flavors to choose from

    How many different combinations of flavors of three scoops of ice cream are possible if it is permissible to make all three scoops the same flavor ?

    The answer is

    all diff + all similar + 2 similar and 3rd diff
    4 choose 3 + 4 choose 1 + 4P2 (the last one is the permutation)

    Can any one justify 4P2 ?
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  2. #2
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    Quote Originally Posted by Hitman6267 View Post
    we have 4 distinct ice cream flavors to choose from

    How many different combinations of flavors of three scoops of ice cream are possible if it is permissible to make all three scoops the same flavor ?

    The answer is

    all diff + all similar + 2 similar and 3rd diff
    4 choose 3 + 4 choose 1 + 4P2 (the last one is the permutation)

    Can any one justify 4P2 ?
    Yeah 4P2 works, although I just thought of it as \displaystyle 2\cdot\binom{4}{2} since once you choose the two flavours, you have to choose which one is the double scoop, meaning multiply by two.

    In case you don't feel I answered your question.. Think of arranging the two flavours in a line, like (vanilla, strawberry) where the first slots means double scoop and the second slot means single scoop. So it's an ordered 2-subset, because you are permuting for example {vanilla, strawberry}, hence 4P2.
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