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Problem of the Day #112: Albert’s Sequence of Block July 9, 2011

Posted by Billy in : potd , trackback

Albert has a large collection of blocks of lengths $1$, $2$, $3$, and $4$. Albert wishes to arrange these blocks in a line such that the length of his line of blocks is $15$. However, the blocks that have length equal to a prime number are negatively charged, so they repel one another and cannot be placed adjacent to another block whose length is prime. Also, Albert can only place blocks immediately to the right of other blocks. If Albert chooses a possible configuration of blocks at random whose length is $15$, the probability that the rightmost block has a prime length can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime integers. Compute $a+b$.

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