JCCDQBHWHCB009(ii) Combinatorics : Case-by-case Analysis (Addition Principle)

Suggested Approach and Solution:-

     For questions with special conditions, a good approach is to consider to the special conditions first.  It matters whether the last digit is a ‘3’ or not.  This is a complication that is handled by a Case-by-case Analysis and then totaling up the number of possibilities.

Case 1: 3 is the last digit

 

If 3 is the last digit, we have 1 choice (i.e. Hobson’s choice) for the last digit.  The first digit must be either a 1 or a 5 i.e. 2 choices.  The remaining 5 digits can be filled in  ⁵P₅ = 5!

= 5 x 4 x 3 x 2 x 1 = 120  ways.

Number of ways for this case = 2 x 5! = 240 ways.

Case 2: 3 is not the last digit

 

In this case, the last digit must be either a 1 or a 5 i.e. 2 choices.  Because these digits only occur once, the first digit will definitely be different from the last digit.  So we need not worry about the first-digit condition as it is automatically satisfied.  The front 6 digits can be filled in

          ^6! / _2! = 360

We divide by 2! because the digit 3 is definitely repeated.

Number of ways for this case = 2 x 360 = 720 ways.

The total number of ways = 240 + 720 = 960.