Designated Survivor is based on the idea that the entire US government was killed in a terrorist attack, save for the "designated survivor", Tom Kirkman, who was off-site at the time. The line of succession for president was followed and Kirkman, Secretary of Housing and Urban Development (12th in line), became the president.
However, the show early on has the problem that he does not have any cabinet members since they were all killed in the terrorist attack. But why can't they follow the line of succession for each of those?
For instance, the Secretary of State has its own line of succession. While I'm not sure who all would attend a president's state of the union, I'm quite certain that not all of the United Nations representatives (10th in line) or the ambassadors to six different countries (11th in line) would be there, so they would have survived. Why didn't people like these become cabinet members?