First, we need estimate, how many such 3-digit numbers between 100 and 999 (inclusive) do exist
such that the digits are not repeated.


This maximal possible amount of such numbers is 9*8*7 = 504.

    (any of 9 digits from 1 to 9 in the 1-st position;
     any of 8 remaining digits in the   2-nd position; and
     any of 7 remaining digits in the   3-rd position).



(a)  If 2000 such numbers are printed out, at least how many of them will be identical ?/.


     Now, I state that of 2000 such printed numbers, there are at least 3 (three) identical numbers.


     Indeed, if any printed number is repeating less than 3 times, than the total amount of printed instances     
     would be not more than 3*504 = 1512;  but in reality, we have 2000 (!) printed numbers.

     Thus the ANSWER to question (a) is 3.



(b)  At least how many of the numbers should be printed, so that at least 3 of them will be identical ?


     From the logic of my solutions to (a), it is clear, that at least 3*504+1 = 1513 such numbers should be printed, 
     so that at least 3 of them will be identical. 


     Thus the ANSWER to question (b) is 3*504+1 = 1513.



(c)  At least how many of the numbers should be printed, so that the number 243 is printed at least 6 times ?


     This question is posed INCORRECTLY:  infinitely many such numbers can be printed, NO ONE of which is the number 243.