1.

First situation:

Put the two students together. This means there are two vacant 'slots' for 10 remaining students.

Second situation:

None of the two concerned students are in the delegation, which means, you have 4 vacant slots for 10 students.

2. For this one, let A B C D E F G H I J K L be the students.

A and B want to be together, and C and D refuse to be together in the delegation.

Possible situations:

(i) A and B are in the delegation with any of E to L students.

(ii) A and B are in the delegation with C and any other student besides D.

(iii) A and B are in the delegation with D and any other student besides C.

(iv) Any of E through L students are in the delegation.

(v) C and any other student from E to L are in the delegation.

(vi) D and any other student from E to L are in the delegation.

Can you work those out?