Lesson A confusing combinatorial problem on repeating digits in numbers

A confusing combinatorial problem on repeating digits in numbers

Problem 1

           Solve step by step


(a)  Let's consider more simple problem first:

         Given three different symbols A, B and C as an alphabet, 
         how many different 5-symbol words can be formed
         such that symbol A is used twice, symbol B is used twice 
         and symbol C is used once?


         For this problem, we can select a position for A in   = 10 ways;
                                         a position for B in   = 3 ways;
                                         then C occupies the remaining 5th position, at no choice.

         So, for this problems, there are 10*3 = 30 different words.

         This outcome is the same as if you use well known formula for repeating letters

              =  = 120/4}}} = 30.



(b)  Now, returning to the given problem, we can assign any of 5 possible digit to A;
                                                        any of remaining 4 digits to B,
                                                    and any of remaining 3 digits to C.
  

     It gives 5*4*3 = 60 different choices for assigning; but for us, the pairs 
     (A=2,B=3) and (B=2,A=3) lead to identical undistinguishable numbers.


     Therefore, this number 60 we should divide by 2 and take 60/2 = 30.



(c)  Now the final step to complete the solution is to multiply 30 from (a) by 30 from (b)

     getting the  ANSWER  30 * 30 = 900.