Distinct die faces seen before the first repeat

A fair six-sided die is thrown repeatedly until some face appears for the second time. What is the expected number of different faces seen before that repetition happens?

$899/324$. Write $X$ for the count and use $E[X]=\sum_k P(X\ge k)$. Having $k$ distinct faces requires the first $k$ throws to be all different, with probability $\prod_{j<k}\tfrac{6-j}{6}$. Summing $1+\tfrac56+\tfrac{20}{36}+\tfrac{60}{216}+\tfrac{120}{1296}+\tfrac{120}{7776}$ gives $\tfrac{899}{324}$.