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Math Help - Combination problem

  1. #1
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    Thumbs up Combination problem

    I need a little help with this problem:-

    A committee is formed consisting of one representative from each of the 50 states in the United States, where the representative from a state is either the governor or one of the two senators from that state. How many ways are there to form this committee?

    I tried using the formula but am not getting the correct answer.
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  2. #2
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    Quote Originally Posted by sllcapri21 View Post
    I need a little help with this problem:-

    A committee is formed consisting of one representative from each of the 50 states in the United States, where the representative from a state is either the governor or one of the two senators from that state. How many ways are there to form this committee?

    I tried using the formula but am not getting the correct answer.
    Dear sllcapri,

    Can you tell me how many ways are there to choose a representative from one state?
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  3. #3
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    2, either a senator or governor
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  4. #4
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    Quote Originally Posted by sllcapri21 View Post
    2, either a senator or governor
    Dear sllcapri,

    But you can choose the governor or one of the two senators. If we think that the senators are distinct from one another (it is not given in the problem so we will have to assume.) how many ways??
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  5. #5
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    In that case it would be three.

    Thanks for responding by the way.
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  6. #6
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    Quote Originally Posted by sllcapri21 View Post
    In that case it would be three.

    Thanks for responding by the way.
    Dear sllcapri21,

    Correct. Now there are 50 states is'nt?? So for the first state we could assign one of the three members. Similarly for the second, third, forth, fifth,...........and fiftieth. So using the "Basic principle of counting" can you give me an idea about the ways in which the commitee could be formed.
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  7. #7
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    It could consist of all senators, all governors or a mixture of both.
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  8. #8
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    Quote Originally Posted by sllcapri21 View Post
    It could consist of all senators, all governors or a mixture of both.
    Dear sllcapri21,

    Well thats an idea...But do you know about the "Basic principle of counting"?? If you don't please refer http://en.wikipedia.org/wiki/Rule_of_product.(Just read the first few paragraphs)
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  9. #9
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    It's a bit confusing to me.
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  10. #10
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    Quote Originally Posted by sllcapri21 View Post
    It's a bit confusing to me.
    Dear sllcapri,

    Possibly you are confused by the complexity of the definition. But if I explain it simply, it means that if you have two events which could be done in n_{1} and n_{2} ways respectively,the total number of ways both events could be done is n_{1}\times{n_{2}}.

    Example:Suppose you toss a coin and a dice,

    Number of outcomes from the coin=2

    Number of outcomes from the dice=6

    Therefore from the "Basic principle of counting",

    Number of total outcomes= 2\times{6}=12

    Did you get the idea??

    You could write the sample space and verify...

    S={Heads and 1,Heads and 2,Heads and 3,Heads and 4,Heads and 5,Heads and 6,Tails and 1,Tails and 2,Tails and 3,Tails and 4,Tails and 5,Tails and 6}
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  11. #11
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    So would it then be 3x50
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  12. #12
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    Quote Originally Posted by sllcapri21 View Post
    So would it then be 3x50
    Ah! I suppose you have'nt undestand it fully. Please reread and try again. Then check the answer.

    Spoiler:
    Retuning to your original question,

    Number of ways to select the member for the first state=3

    Number of ways to select the member for the second state=3
    . . .
    . . .
    . . .
    . . .
    . . .
    . . .
    . . .
    . . .
    Number of ways to select the member for the fiftieth state=3

    Therefore by the "Basic princilpe of counting" total number of ways= 3\times{3}\times{3}............\times{3}~(fifty ~3s')=3^{50}


    Did you understand??
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  13. #13
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    Oh, I see. Not 3x50 but 3xitself, 50 times. Thanks so much. I will continue to practice these types of problems.
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