4. the majority of the court believed that compulsory exclusion of large groups of citizens from their homes was okay in what situation?

I think what you are referring to is the Supreme Court case of Korematsu v. United States 

The case was about a 20-something year old Japanese American who refused to relocate to the detention camp that the military had ordered him to (because of the Pearl Harbor attack). 

The Supreme Court ruled that "...compulsory exclusion of large groups of citizens from their homes, except under circumstances of direct emergency and peril, is inconsistent with our basic governmental institutions. But when under conditions of modern warfare our shores are threatened by hostile forces, the power to protect must be commensurate with the threatened danger". 

The court ruled that compulsory exclusions is okay in cases of MILITARY NECESSITY. 

Their reasoning was that citizenship has it privileges but also has it responsibilities so when the military was rounding up all the citizens of Japanese descent, they had the duty to obey by way of the responsibilities of American citizenship.