You will see phrase “in care of name” in Part 1. Information about you, the conditional resident, of Form I-751, namely in mailing and physical address sections.
“In care of” in form i-751, shortly c/o, applies only when you want someone else to receive mail on your behalf. For example, if you are using your parents’ address for postal deliveries, but you do not actually live with them; or you live in your relatives’ residences and do not have your own residence; or if you are using a lawyer, an immigration assistance organization to file the documents, or just someone who is better prepared to receive mail than you. Any correspondence can be sent to this address instead of your actual address.
“In care of” only applies if the address you are using is not your address, but someone else’s. So if you’re using your own permanent address, you don’t need to fill that part in.