The probability that in a random arrangement of the letters of the...

The total number of arrangements of the letters of the word 'UNIVERSITY' is $\frac{10!}{2!}$ as there are two I's. Regarding 2I's as one letter, number of ways of arrangement in which both I's are together $=9$ ! So, the number of ways in which 2 I's are not together $=\frac{10!}{2!}-9!$ Hence, the required probability $=\frac{\frac{10!}{2!}-9!}{\frac{10!}{2!}}=1-\frac{2!9!}{10!}=\frac{4}{5}$.