The correct answer is option 3: 1 and 3 only.

Option 1 states that the Security Council of the United Nations has 15 members, which is correct. The Security Council consists of five permanent members - China, France, Russia, the United Kingdom, and the United States - and ten non-permanent members.

Option 2 states that non-permanent members are elected by the General Assembly for a five-year period. This statement is incorrect. Non-permanent members of the Security Council are actually elected by the General Assembly for a two-year period. The General Assembly takes into account equitable geographical distribution and rotation among member states when electing non-permanent members.

Option 3 states that retiring members are not eligible for immediate re-election, which is correct. This rule ensures that the composition of the Security Council remains diverse and allows for new perspectives and opinions.

In summary, the correct statements are that the Security Council has 15 members and retiring members are not eligible for immediate re-election. The incorrect statement is that non-permanent members are elected for a five-year period.