The names of North Carolina: Four-Year Colleges and Universities as your cousin wants to take admission in it only are as follows:

Appalachian State University

Barber-Scotia College

Belmont Abbey College

Campbell University

Catawba College

Davidson College

Duke University

East Carolina University

Gardner-Webb University

High Point University

Johnson and Wales University

Lee University

Lenoir-Rhyne College

Livingstone College

Meredith College

Montreat College

North Carolina State University

Peace College

Pfeiffer University

Queens University of Charlotte

Strayer University

University of North Carolina

University of North Carolina Charlotte

Wake Forest University

Western Carolina University

Wingate University

Winston-Salem State University