The formulation of this post seems to be tricky, so we need re-formulate it in clean, clear and simple form.


The 4 papers can be checked by any team of 4 teachers; by any team of 3 teachers; by any team of 2 teachers; 
and, finally, by any one of 7 teachers.


So we need to calculate the sum 


    sum =  +  +  +      (which is the number of elements in the space of events)


and then calculate the probability as the ratio  P = ,   where   is the number of favorable events.


In this way we will get the answer.


     =  = 35;

     =  = 35;

     =    = 21;

               = 7.


So,  sum = 35 + 35 + 21 + 7 = 98.


Therefore,  P =  =  = .


Since we are given that   P =  = ,  it implies


    k =  =  =  = .    ANSWER


ANSWER.  k = .