Friday, April 6, 2012
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.