Question 60
Teresa the bunny has a fair 8-sided die. Seven of its sides have fixed labels $1, 2, \ldots, 7$, and the label on the eighth side can be changed and begins as $1$. She rolls it several times, until each of $1, 2, \ldots, 7$ appears at least once. After each roll, if $k$ is the smallest positive integer that she has not rolled so far, she relabels the eighth side with $k$. The probability that $7$ is the last number she rolls is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.