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Math Help - ordered and unordered partitions

  1. #1
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    ordered and unordered partitions

    In how many ways can 10 students be divided into three teams, one containing 4 students and the others 3?

    Method 1 makes sense, in which the answer is \frac{10!}{4!3!3!}\cdot\frac{1}{2}.

    But method 2 bothers me, where the answer is \binom{10}{4}\binom{5}{2}.

    I can understand choosing 4 out of 10 the first time, but what is the logical explanation for choosing 2 out 5 the second time? This does not seem very consistent.
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  2. #2
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    Quote Originally Posted by novice View Post
    In how many ways can 10 students be divided into three teams, one containing 4 students and the others 3?

    Method 1 makes sense, in which the answer is \frac{10!}{4!3!3!}\cdot\frac{1}{2}.

    But method 2 bothers me, where the answer is \binom{10}{4}\binom{5}{2}.

    I can understand choosing 4 out of 10 the first time, but what is the logical explanation for choosing 2 out 5 the second time? This does not seem very consistent.
    Imagine you're the coach for this squad. You pick four of the ten students for the first team ( \textstyle{10\choose4} ways of doing that). You then designate one of the other six students (doesn't matter who it is) to be a team captain, with instructions to choose two of the remaining five students to complete his/her team ( \textstyle{5\choose2} ways of doing that). The three left-over students form the third team, of course.
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  3. #3
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    Quote Originally Posted by Opalg View Post
    Imagine you're the coach for this squad. You pick four of the ten students for the first team ( \textstyle{10\choose4} ways of doing that). You then designate one of the other six students (doesn't matter who it is) to be a team captain, with instructions to choose two of the remaining five students to complete his/her team ( \textstyle{5\choose2} ways of doing that). The three left-over students form the third team, of course.
    Thank you, Opalg, for taking time answering my question. Your explanation makes a lot of sense. We need more of your kind to write math books.
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