ShadowMourne:
var result = db.Students
            .Join(db.Students, s1 => s1.FacultyID, s2 => s2.FacultyID, (s1, s2) => new
            {
                LastName1 = s1.LastName,
                LastName2 = s2.LastName,
                ID1 = s1.ID,
                ID2 = s2.ID
            }).Where(y => y.ID1 != y.ID2 &&
              !db.Students
                    .Join(db.Enrollments, s => s.ID, e => e.StudentID, (s, e) => new
                    {
                        e.CourseID,
                        s.LastName,
                        s.ID
                    })
                    .Join(db.Courses, c => c.CourseID, e => e.CourseID, (c, e) => new
                    {
                        c.CourseID,
                        c.LastName,
                        c.ID
                    }).Where(x => x.ID == y.ID1)
              .Except(
                    db.Students
                    .Join(db.Enrollments, s => s.ID, e => e.StudentID, (s, e) => new
                    {
                        e.CourseID,
                        s.LastName,
                        s.ID
                    })
                    .Join(db.Courses, c => c.CourseID, e => e.CourseID, (c, e) => new
                    {
                        c.CourseID,
                        c.LastName,
                        c.ID
                    }).Where(x => x.ID == y.ID2)
                    ).Any()
                    &&
                    !db.Students
                    .Join(db.Enrollments, s => s.ID, e => e.StudentID, (s, e) => new
                    {
                        e.CourseID,
                        s.LastName,
                        s.ID
                    })
                    .Join(db.Courses, c => c.CourseID, e => e.CourseID, (c, e) => new
                    {
                        c.CourseID,
                        c.LastName,
                        c.ID
                    }).Where(x => x.ID == y.ID2)
              .Except(
                    db.Students
                    .Join(db.Enrollments, s => s.ID, e => e.StudentID, (s, e) => new
                    {
                        e.CourseID,
                        s.LastName,
                        s.ID
                    })
                    .Join(db.Courses, c => c.CourseID, e => e.CourseID, (c, e) => new
                    {
                        c.CourseID,
                        c.LastName,
                        c.ID
                    }).Where(x => x.ID == y.ID1)
                    ).Any()
                    )
                    .Select(x => new StudentViewModel
                    {
                        LastName1 = x.LastName1,
                        LastName2 = x.LastName2
                    })
                    .ToList();
            result.ForEach(x => resultString += string.Concat(x.LastName1, x.LastName2));