EXAMPLE 3.1.15

Erasmus is trying to guess the combination to his combination lock. The "combination" is a sequence of three numbers, where the numbers range from 1 to 12, with no numbers repeated. How many different "combinations" are possible if he knows that the last number in the combination is either 1 or 11?

A. 264                         B. 1320           C. 220                         D. 288

 

 

EXAMPLE 1.4.15 solution

 

Choosing a three-number ńcombinationî having no repeated numbers requires that we make three dependent decisions.  One of these decisions, however, has a special condition attached to it (the third number must be either 1 or 11).  When using the Fundamental Counting Principle in a situation involving dependent decisions, if one decision has a special condition, that decision must be treated first, because the special condition overrides the other decisions.  For example, that fact that the third number must be 1 or 11 means that it is impossible for the ńcombinationî to simultaneously have 1 for the first number and 11 for the second number (since then there would be nothing left for the third number).

 

Three dependent decisions:

1.  Choose third number (two options);

2.  Choose first number (11 options);

3.  Choose second number (10 options).

 

According to the Fundamental Counting Principle the number of possible outcomes is

(2) times (11) times (10) = 220.

 

The correct choice is C.