I’m still reading the book used by our department for “Topics in Finite Mathematics” and I found this interesting little snippet from the introduction to the section entitled “The Multiplication Principle”.

Here are some examples of counting problems that are unlike any we have seen up to this point, but that we should be able to answer after studying this section:

How many different Social Security numbers are possible?

How many different telephone numbers can be given the area code 870?

In how many ways can a president and a vice president be selected from among 12 people?

How many ways can the sum of eight be rolled on a pair of dice?

Here is my question for everyone: Which of the above four questions is the most different from the others? If you like, you might rephrase this as: Which of the above four questions doesn’t belong?

My answer differed from my wife’s answer and I think both are valid (but I won’t say what they were yet). I’d love to hear some other responses to this. If you happen to guess the same one I chose—for the same reason—then you win the grand prize for making me feel less alone in the universe. Sadly, the grand prize is merely reciprocity.

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The dice one is the only one not associated with government (was that your wife’s response?). The dice one also seems the trickiest in terms of the math. The first two seem like they’d have some sort of rules associated with them (can anyone have SS# 000-00-0000? can you have a phone number that starts with 0 or 1 or 911?)

My guess for your answer is: Pres and VP are not as “random” as the others.

Comment by CalcDave — August 29, 2010 @ 7:55 pm |

The dice is different. It must be split into cases to solve (“what if the first die is a one?…what if the first die is a two?…). Then you sum the cases together, thereby giving it an additive flavor that the other problems lack.

Comment by bretbenesh — August 29, 2010 @ 9:25 pm |

The first two problems are very similar (multiply independent choices, possibly excluding some illegal combinations), so you must feel one of the latter 2 is different. The P/VP choosing is another multiplication (12*11), but the dice one is a simple count or subtraction. So my vote would be for the dice problem being the different one.

Comment by gasstationwithoutpumps — August 30, 2010 @ 10:11 am |

CalcDave gets the prize for being closest to my answer. But I love the other two responses because they bring up something I hadn’t thought about: the dice problem isn’t typically solved using the multiplication principle (the subject of the section). Furthermore, despite the promise that the fourth question is a problem “that we should be able to answer after studying this section,” they never talk about that kind of problem in that section (the text of the exercises). What a rip-off!

My answer was the third because of the four problems, it is the only one that I would have no clue how to motivate to a class. I mean, when would you realistically need to know the answer to that question? Sure, some people need to consider how many SSN’s there should be, the area code problem certainly shows up, and a compulsive gambler—or Settlers of Catan nut—can tell you the importance of the dice problem. But when would you care how many ways you could choose P/VP from 12 people?

Comment by Adam Glesser — August 30, 2010 @ 8:14 pm |