SOLUTION: There are 4 students attending a test and 7 teachers available for checking the tests. Each student's test is checked by exactly one of these teachers, with each teacher being equa

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 = .