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Government-Issued Identity Crisis?

Date: 01/26/2011 at 18:51:43
From: Mark
Subject: Permutations of social secirity numbers

I have a couple questions about Social Security Numbers (SSNs).

I'm going to assume that they all take the same format as my own. Namely,
they are all nine digits long, the pool of numbers for each digit is ten,
and SSNs are broken up into three blocks:

   XXX-XX-XXXX

Now, when I run permutations on the triple block, and then the double, and
then the 4-tuple, I come up with ~325,000,000 possible values for SSNs,
where each is a different number.

Is the way I handled my permutation calculations correct? 

If so, and there are ~300,000,000+ people in the US, does that mean
that -- unless we change the format -- we will run out of SSNs soon?

Philosophically, is my conclusion correct?

Thank you.



Date: 01/26/2011 at 19:31:54
From: Doctor Wallace
Subject: Re: Permutations of social secirity numbers

Dear Mark,

It appears you have multiplied 10P3 times 10P2 times 10P4 to arrive at
your estimation of 325 million permutations. The exact product would be
326,592,000.

Still, if this were the maximum number of permutations of a 9-digit SSN,
then your conclusion would be correct: we would be on the verge of
running out of numbers.

However, your calculation has placed an artificial restriction on the
problem. You have calculated the number of permutations if no digit is
permitted to be reused within each block of numbers.

That is to say, your calculation would not consider for example 
112-33-6671 as a valid SSN.

Social Security Numbers are grouped into three sets because they were
assigned from geographical regions. You can find out more about this
coding here:

   http://www.ssa.gov/history/ssn/geocard.html 
   
So the calculation you're trying to perform would strip away that coded
meaning.

Now, if we just consider the SSN as a simple 9-digit number, there would
be 10^9 possible permutations: just count from 000-00-0000 all the way to
999-99-9999.  This is because any digit (0 - 9) can occur in any of the
nine places, making for ...

   10*10*10*10*10*10*10*10*10 = 10^9
   
... or 1 billion possibilities.

The Social Security Administration's website notes that they have assigned
approximately 420 million different numbers, and that they assign about
5.5 million new numbers every year. Even with a billion numbers possible
under the current numbering system, eventually they will run out and have
to change the system.

Let's estimate just how soon this will happen, starting with the number of
SSNs as yet unassigned:

   1 billion - 420 million = 580 million

At 5.5 million per year,

   580/5.5 = 105.5 years

So they're good for at least another century or so before having to change
the system.

Learn more here:

   http://www.ssa.gov/history/hfaq.html 

Don't hesitate to write again if you need further help with this or
another question.

Thanks for writing to Dr. Math!

- Doctor Wallace, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 01/27/2011 at 16:40:17
From: Mark
Subject: Thank you (Permutations of social secirity numbers)

Thank you, Dr. Wallace. Your explanations were clear, concise, quick --
and much appreciated!

This is the first time I've contacted the Math Forum. But since I'm very
interested in math, statistics, and physics, I'm sure I will do it again
when I'm stuck.

Thanks again.

Mark Sabla
Associated Topics:
Elementary Large Numbers
High School Permutations and Combinations

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