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How many arrangements of five 0's and six 1's are there with no consecutive 1's?

I found this question in a book named Challenge and thrill of pre college mathematics ; My solving goes like fixing the places of 0's and filling 1 Between the 6 places - "0_0_0_0_0" But i dunno how to obtain the value , i would be grateful for any hint or solutions.

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  • $\begingroup$ Sry but i am unable to add _ at the extreme ends of my solution $\endgroup$
    – Unique_1
    Commented Oct 26, 2022 at 16:42
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    $\begingroup$ Place the $6$ ones with gaps between them. Each gap between two ones must have at least one zero, but that requires $5$ zeroes, so there is only the one way. $\endgroup$
    – lulu
    Commented Oct 26, 2022 at 16:45

1 Answer 1

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10101010101 is the only arrangement of 6 1's, 5 zeros, and no consecutive 1's. Essentially every 1, must be at the beginning, the end, or be in a ...010... pattern.

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