At a certain university, 3 out of every 5 students live in an on-campus dormitory. If one out of every 10 students who

Students in on-campus dormitory = 3 out of 5 = 3/5
Students that are not in on-campus lives in a university-owned apartment = 1 out of 10 = 1/10
Let total students = x
$$\frac{3}{5}x=on\ campus\ and\ remaining\ student$$
$$=\frac{1x}{1}-\frac{3x}{5}=\frac{5x-3x}{5}=\frac{2x}{5}$$
From the remaining 2x/5, 1/10 students who are not on campus dormitory are in the university-owned apartment.
$$Therefore,\ \frac{1}{10}\cdot\frac{2x}{5}=\frac{1}{25}x\ live\ in\ the\ university-owned\ apartment.$$
Students who do not live on campus and do not live in a university-owned apartment will be;
$$=\frac{2}{5}x-\frac{1}{25}x=\frac{10x-x}{25}=\frac{9x}{25}$$
As a fraction of total, 9/25 of the total students lived in neither on-campus dormitory nor university-owned apartments. Hence, the correct answer is option C.