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Math Help - subsets question

  1. #1
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    subsets question

    Hey iv been stuck on this question for a while now and am convinced its just because im missing something obvious so was wondering if anyone can give me some help.

    What proportion of 6-element subsets of {1, 2, . . . , 49} contain at least two consecutive integers? What proportion of 7-element subsets of {1, 2, . . . , 49} contain at least two con-secutive integers?

    thanks
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  2. #2
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    Quote Originally Posted by CHAYNES View Post
    What proportion of 6-element subsets of {1, 2, . . . , 49} contain at least two consecutive integers? What proportion of 7-element subsets of {1, 2, . . . , 49} contain at least two consecutive integers?
    How many bit-strings of length 49 contain exactly 6 ones but no two ones are consecutive?
    The answer is \binom{44}{6}. The 6 ones are separated by at least one zero.
    This the number of 6-element subsets of {1, 2, . . ., 49} containing no consecutive integers.
    The answer to the question is 2^{49}-\binom{44}{6}.
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