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Math Help - Counting

  1. #1
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    Counting

    hello

    I have a test in a few days.
    Could you help me with these two counting problems:

    1)
    a) what is the number of series (length 19)in which there are 8 0's and 11
    1's?
    answer: easy C(11,8) the problem is with part b:

    b) among the series above, what is the number of series in which in any prefix with length at least 3 the number of 1's is larger than the number of 0's by 3?

    well I really dont know how to solve this thing..help would be appreciated.

    2) what is the number of length 4 cycles in the complete bipartite graph K(6,6)?

    thanks for your help guys
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  2. #2
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    Quote Originally Posted by parallel View Post
    1) a) what is the number of series (length 19)in which there are 8 0's and 11 1's?
    answer: easy C(11,8) WRONG the answer is C(19,8).

    b) among the series above, what is the number of series in which in any prefix with length at least 3 the number of 1's is larger than the number of 0's by 3?
    I cannot decipher what that means. What does “any prefix with length at least 3” mean?
    Please correct.
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  3. #3
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    of course it's C(19,8)

    I'll give you an example:

    let's say the prefix is of length 3
    so the only combination is: 111 (it's the only one)

    let's look at the prefix of length 4: the way I see it ,there is no way to build this kind of prefix,becuase there can't be a combination in which the number of 1's is larger than the number of 0's by 3

    for the prefix of length 5 the combinations are:
    11110,10111...etc

    we need to check each prefix, until we get to the prefix of length 19

    is it clear now?
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