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Lin the Racoon's Debut

Author: mathforces
Problem has been solved: 12 times

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Lin the Racoon compiled a sequence of $ 100 $ digits in which exactly $ 50 $ digits $ 0 $ and $ 50 $ digits $ 1 $. At the same time, for any positive integer $ k $ from $ 1 $ to $ 100 $, among the first $ k $ digits of Lin the Racoon's sequence, the digit $ 0 $ does not occur more often than the digit $ 1 $. And also for any $ i = 44, 45, ..., 57 $, if the $ i $ -th digit of its sequence is $ 1 $, then the $ i + 1 $ -th digit of this sequence is also $ 1 $. Let $x$ be number of possible sequences that Lin the Racoon could make up. Find the reminder when $x$ is divided by $10^7$.





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