Can someone please check my answers? If any are wrong I'll post my working out.
$(a)$ I roll five standard $6$-sided dice, and write down the five values shown. Calculate the probability of each of the following events.
$(i)$ All five dice show the same value. $\frac{6}{6^5}$
$(ii)$ The dice give five distinct values. $\frac{6P5}{6^5} = \frac{6!}{6^5}$
$(iii)$ The dice give five consecutive values (not necessarily in the order rolled). Only possibilities are ordering $\{\,1,2,3,4,5\,\}$ and $\{\,2,3,4,5,6\,\}$. Ordering is 5! for each, so $\frac{5! \cdot 2}{6^5}$
$(iv)$ Three dice show $x$ and two dice show $y$, for some integers $x\neq y$. There are $6$ possibilities for $x$ and $5$ possibilities for $y$. To order $(x,x,x,y,y)$ it is $\frac{5!}{2!\cdot3!}$. So, $\frac{30 \cdot \frac{5!}{2!\cdot3!}}{6^5}$