The first student can be chosen as any the 11 students.
The second student can be chosen as any of the 10 remaining unchosen students.
The third student can be chosen as any of the 9 remaining unchosen students.
The fourth student can be chosen as any of the 8 remaining unchosen students.
The fifth student can be chosen as any of the 7 remaining unchosen students.
The sixth student can be chosen as any of the 6 remaining unchosen students.
The seventh student can be chosen as any of the 5 remaining unchosen students.
The eighth student can be chosen as any of the 4 remaining unchosen students.
The ninth student can be chosen as any of the 3 remaining unchosen students.
The tenth student can be chosen as either of the 2 remaining unchosen students.
The eleventh student must be chosen as the 1 remaining unchosen student.

That's 11∙10∙9∙8∙7∙6∙5∙4∙3∙2∙1 = 11! = 11P11 = 39916800 ways.

The first student can be chosen as any the 11 students.
The second student can be chosen as any of the 10 remaining unchosen students.

That's 11∙10 = 11P2 = 110 ways.

The hall monitor can be chosen as any the 11 students.
The board cleaner can be chosen as any of the 10 remaining unchosen students.
The lunch hauler can be chosen as any of the 9 remaining unchosen students.

That's 11∙10∙9 = 11P3 = 990 ways.

Edwin

    Two students to serve on the 3rd grade council

    can be chosen by    =  = 55 ways.

    The order in the pair does not matter in this case.