Rolling a die until a value repeats

You roll a fair six-sided die over and over, stopping on the first roll that shows a value seen earlier. Let $r$ be the total number of rolls and $p_r=\mathbb{P}(r)$.

(a) Find $p_3$.

(b) Find $p_1+p_2+\cdots+p_{10}$.

(c) Find the probability that $r=5$ given that the game lasts at least three rolls.

(a) $5/18$. (b) $1$. (c) $2/9$. For (a), the second roll must be new ($\tfrac56$) and the third a repeat ($\tfrac26$). For (b), only six distinct values exist, so a repeat must occur by roll $7$ and all probability lies within $r\le10$. For (c), $p_5=\tfrac56\cdot\tfrac46\cdot\tfrac36\cdot\tfrac46=\tfrac5{27}$, divided by $\mathbb{P}(r\ge3)=\tfrac56$.