To solve this problem, we need to determine at which stage of HIV infection symptoms of AIDS typically appear. Let's analyze each option:$\newline \bullet$Option 1: Within 15 days of sexual contact with an infected person.$\newline$ - This is incorrect because symptoms of AIDS do not appear so soon after infection. The initial stage is often asymptomatic or may present with flu-like symptoms.$\newline \bullet$Option 2: When the infected retrovirus enters host cells.$\newline$ - This is incorrect because the entry of the virus into host cells marks the beginning of the infection, not the onset of AIDS symptoms.$\newline \bullet$Option 3: When HIV damages a large number of helper T-Lymphocytes.$\newline$ - This is correct. AIDS symptoms appear when the immune system is severely compromised due to the significant depletion of helper T-Lymphocytes (CD4 cells).$\newline \bullet$Option 4: When the viral DNA is produced by reverse transcriptase.$\newline$ - This is incorrect because the production of viral DNA is an early step in the HIV replication process, not directly related to the onset of AIDS symptoms.$\newline$Therefore, the correct answer is Option 3: When HIV damages a large number of helper T-Lymphocytes.