If you get rejected from the Barrett Honors college at Arizona State University can no longer be considered for general admission into school?

No, being rejected from the Barrett Honors College at Arizona State University does not mean you can no longer be considered for general admission into the school. The Barrett Honors College is a selective honors program within ASU, and admission to the program is separate from admission to the university itself. If you are not admitted to the Barrett Honors College, you can still be admitted to ASU as a general student.